Geometric Measure Theory in Banach Spaces

CHAPTER

35

Geometric Measure Theory in Banach Spaces

David Preiss Department of Mathematics, University College London, London WC1E 6BT, UK E-mail: [email protected]

Contents 1. F i n i t e - d i m e n s i o n a l g e o m e t r i c m e a s u r e theory in i n f i n i t e - d i m e n s i o n a l situations . . . . . . . . . . . . . . 1.1. Rectifiability and density 1.2. Currents

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2. I n f i n i t e - d i m e n s i o n a l g e o m e t r i c m e a s u r e theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.1. Differentiable m e a s u r e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.2. Surface m e a s u r e s

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2.3. M e a s u r e s and balls

2.4. Differentiation t h e o r e m s for G a u s s i a n m e a s u r e s

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3. Exceptional sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4. Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.1. C o n v e x functions 4.2. Lipschitz functions References

HANDBOOK

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OF THE GEOMETRY OF BANACH SPACES, VOL. 2

E d i t e d by W i l l i a m B. J o h n s o n and J o r a m L i n d e n s t r a u s s 9 2003 Elsevier S c i e n c e B.V. All rights r e s e r v e d 1519

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We will consider the question to what extent the classical relations between measure, derivative and geometry carry over to infinite-dimensional Banach spaces. Such relations may be strangely distorted, like the seemingly simple question of recovering the R a d o n Nikod2~m derivative by the limit of ratios of measures of balls, or answers may not be known, even in basic cases such as existence of c o m m o n points of Fr6chet differentiability of finitely many real-valued Lipschitz functions on a separable Hilbert space. Our goal is to describe some basic notions and results; these notes should be considered as an invitation to the subject and not as a survey of the subject, since many important concepts have necessarily been left out. We first visit two themes which are essentially finite-dimensional even though the surrounding space is infinite-dimensional, the problem of relations between rectifiability and density in general metric spaces and the recently developed theory of currents in metric spaces. In both cases, the structure of a Banach space is not essential for the setting, but it may always be assumed and it was needed to prove some of the deep results. For practically all problems of infinite-dimensional geometric measure theory, the most important difference between finite-dimensional and infinite-dimensional Banach spaces is due to non-existence of a reasonably finite, translationally invariant measure in the latter case: since any ball B ( x , r) contains infinitely many disjoint balls of radius r / 3 , if lz is a non-zero Borel measure on an infinite-dimensional separable Banach space X a n d the lz-measure o f balls depends only on their radii, then every non-empty open set has infinite measure. In fact, i f / z is a a-finite measure on X, the shifted measure is singular with respect to # for many shifts from X. One way to see this is to assume, as we may, that ~ is finite and the norm is square integrable and consider the C a m e r o n - M a r t i n space oflz, H = {x E X; sup{Ix*(x)l: IIx*llc,_(j~)} < ~ } . The identity from H to X is compact (in fact, it is much better; for example, if X is Hilbert, it is Hilbert-Schmidt), and it is easy to see that the shift of # by any vector not belonging to H is singular with respect to #. Nevertheless, there are measures in infinite-dimensional Banach spaces that have some of the properties normally associated with the Lebesgue measure. In many instances, Gaussian measures have been used as a replacement for the Lebesgue measure. A more general class of such measures is formed by those that are quasi-invariant in a dense set of directions, where a measure is called quasi-invariant in a direction u if its null sets are preserved when shifted by u. We will restrict ourselves to a discussion of the notions of measures differentiable in a direction in 2.1 and to briefly pointing out connections to the possibility of defining surface measures. In 2.3 we visit the amusing results obtained by attempting to understand to what extent exact nai"ve analogues of finite-dimensional results fail in infinite dimensions. In the short Section 3 we give several notions of exceptional sets. Some readers may find it more convenient to read the definitions from this section only after encountering their use in Section 4, where we treat the problem of existence of Gfiteaux or Fr6chet derivative of Lipschitz functions and also briefly consider few more exotic derivatives. Out of the directions that have been omitted one should definitely mention the study of various notions of generalized derivatives (or subdifferentials) which often parallels the development of classical real analysis. As an example, see [11]. For many other problems that could fit into this text as well as a number of results relevant to it see [5]. Although from time to time we mention the case of Gaussian measures (defined as those measures whose every one-dimensional image by a continuous linear functional is Gaussian or Dirac), much

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of the extensive research devoted to them has not been mentioned even though it often has deep connections to problems of geometric measure theory. To avoid possible misunderstandings, we will consider only separable Banach spaces unless non-separability is specifically permitted, and only Borel measures. Normally, measures will be positive and finite. There are, however, important exceptions: signed (i.e., realvalued) measures are appearing, out of necessity, in our considerations of differentiability of measures in 2.1 and the Hausdorff measures are positive but often notoriously infinite.

1. Finite-dimensional geometric measure theory in infinite-dimensional situations Many situations from the classical geometric measure theory (including the notions of fractal geometry such as definition of fractal sets via iterated function systems) can be easily transferred to the case when the ambient space is infinite-dimensional. For example, the k-dimensional Hausdorff measure

7-/k (A) =

1 lim inf c~(k) ~-,o

diam k(Ai)" A C i=!

A i , diam(Ai) < 6 i=1

as well as a number of several other k-dimensional measures (spherical measures, packing measures, etc.) have been studied in arbitrary metric spaces. (The constant factor c~(k) is chosen so that the k-dimensional Hausdorff measure in It~k coincides with the Lebesgue measure; since we will not need this fact here, for the purpose of these notes we may set c~(k) = 1.) The natural setting is often that of metric spaces (and, because these can be embedded into Banach spaces, it usually suffices to consider only these), but with a few notable exceptions, the generality does not bring much new, even though it may help to clarify the assumptions or provide natural proofs. Here we will briefly consider two of the situations in which the interaction with geometry of Banach spaces went much farther.

1.1. Rectifiability and density Much of the development of classical geometric measure theory was driven by attempts to show, under various geometric assumptions on a subset A of IR" of finite k-dimensional measure, that A is k-rectifiable, i.e., that 7~k-almost all of A can be covered by a Lipschitz image of a subset of IRk. (The restriction to the Hausdorff measure and sets of finite measure is not necessary, but it is convenient and suffices for this presentation.) Perhaps the most useful rectifiability criterion, the Besicovitch-Federer projection theorem (see [17, Theorem 3.3.12]), did not seem to have any natural counterpart for infinite-dimensional ambient spaces till one was found in the modern setting of currents (see below). Another useful rectifiability criterion (due, in increasing level of generality, to Besicovitch, Marstrand and Mattila) says that (under the above assumptions) A is k-rectifiable if and only if its k-dimensional density O)k (A, x) = l i m ~ 0 7-[k(A A B(x, r))/(~(k)r k) is equal to one at 7-/k almost every x 6 A. This could well be true in every metric space, although

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it was shown only in the cases when either k -- 1 or the space in question is a subset of a uniformly convex Banach space. (So the simplest unknown case is the two-dimensional measure in ~ . ) The implication saying that rectifiable sets have density one almost everywhere was proved by Kirchheim [27] in full generality. The key behind this is the existence of metric derivative (see 4.2.6). One of the corollaries of this work is that the area formula remains valid in the general situation; in the simpler case of injective Lipschitz mappings this says that the k-dimensional measure of the image can be calculated as an integral of a suitable Jacobian over the domain, where the Jacobian depends just on the metric derivative of the mapping.

1.2. Currents The development of the theory of currents was motivated by the difficulty to prove existence results for higher-dimensional minimal surfaces by classical methods. The basic ideas behind the (finite-dimensional) theory of currents, as found, for example, in [ 17], were similar to those behind the introduction of distributions: a k-dimensional oriented surface gives rise to a linear functional on the space of differential k-forms; among such functionals one chooses (and calls currents) those that still form a (weak*) compact set but already have many features of 'genuine surfaces'. The main notions may be transfered to Hilbert spaces (or even to those Banach spaces in which suitable results on differentiability of Lipschitz mappings hold) without any change, but deeper results appear to be based on concepts that are not available beyond the finite-dimensional situation. We will now describe an important recent development which shows that the theory needs no concept of differentiability and that many strong results remain valid even when the ambient space is infinite-dimensional. It is due to Ambrosio and Kirchheim [2], based on an idea of De Giorgi and, incidentally, does not even use the concept of differential forms. Instead of considering integration over a k-dimensional (smooth) oriented surface S C ~n as a linear functional on k-dimensional differential forms, we will consider it as a (k + 1)-linear functional on the space 79k+l (R n) of (k + l)-tuples of Lipschitz functions ( f , 7/'!, 792 . . . . . ~k). So, in the simplest case of S = [a, b] being an interval on the line (so k = n -- 1), the associated current is T(U, Jr) = fl,,t,l f drr = fl' f ( x ) r r ' ( x ) d x . Somewhat more generally, if g is a Lebesgue integrable function on R" (which can be imagined as the multiplicity of the k = n-dimensional oriented surface S = {x: g(x) ~ 0}), the associated current is

T ( f , tel,

7r2 . . . . .

7gk) -- fS f g drrl A dn'2 A . . . A dJrk

f

f(x)g(x)det(rci(x),rc2(x ) . . . . . 7r~ (x)) dx,

and analogous formulas associate such (k + l)-linear forms to (smooth) k-dimensional surfaces with real multiplicity also in case when k < n. The key observation is that the functionals T in the above examples have the following three properties.

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(a) T is (k + 1)-linear. (b) The restriction of T to the set of (k + l)-tuples with If[ ~ 1 and Lip(re/) ~< 1 is continuous in the topology of uniform convergence of the f ' s and pointwise convergence of the rei's. (c) T is local in the sense that T(f, re~, re2 . . . . . rek) = 0 w h e n e v e r some rei is constant on {x: f ( x ) :/: 0}. For the currents from our examples, (c) expresses the fact that they depend on the derivatives of the rei's. The fact that (b) holds (which is not so obvious) has been recognised (often implicitly) as important in a n u m b e r of other connections. We can now define a k-dimensional current T in a complete metric space E as a functional on 7) k+! (E) satisfying (a)-(c). The key requirement of locality means that T depends in a weak sense on the derivative of the rei's, and it is the main point in showing that currents have the property that transforming the rei's by a Lipschitz m a p p i n g of IKk multiplies the f by the determinant of the transformation. (In particular, currents are anti-symmetric in the rei's.) One m a y therefore use the more suggestive notation T ( f drel A . . . /x drek) for T(f, rel . . . . . rek), although the symbols drei themselves may have no meaning. Currents behave like measures in the first variable; more precisely there is a finite m e a s u r e / z such that T(f, rel, re2 . . . . . :rk) ~ I-Ii L i p ( r e / ) f If[ dtt; the least such measure is called the mass of T and denoted by 11T I[. The total mass of T is defined as M ( T ) = [ITII(E). Standard operations on currents are defined in a natural way. The existence of mass allows one to extend currents to arbitrary b o u n d e d Borel f ; in particular, the restriction of a current to a Borel set B may be defined as (f, re l . . . . . rek) ~ T ( f x n , Jr. . . . . . rek), where XB is the indicator function of B. The push-forward of T by a Lipschitz mapping r : E --+ F is r T ( f , re l . . . . . :rk) = T ( f o 4), re l o r . . . . . rek o r and the bounda~. of a (k + 1)-dimensional current S is 0 T (f, re, . . . . . rek ) = S( 1, f, re, . . . . . rrk). However, boundaries are tricky: 0 T is a functional satisfying (a) and (c), but there is no reason why it should satisfy the continuity requirement (b). Currents for which 0 T satisfies (b) are called normal and are the first of the basic classes of current. Two other concepts arise naturally from the wish to define a notion that should represent generalized surfaces (with integer multiplicity): a k-dimensional current T is rectifiable if its mass is absolutely continuous with respect to the k-dimensional measure on some k-rectifiable set and it is an integer current if the push-forward to R k of any restriction of T to a Borel set is representable by a Lebesgue integrable integer-valued function. It is not known if every k-dimensional current in R k (k ~> 3) is rectifiable (this problem is close to that of describing sets of non-differentiability of Lipschitz mappings in Ii~k) but normal k-dimensional currents in I~k are necessarily rectifiable and in fact correspond exactly to functions of b o u n d e d variation. It follows that, within normal currents in R", the new concepts coincide with the standard ones. These classes of currents admit many natural characterizations similar to those obtained for currents in finite-dimensional spaces. In particular, strong rectifiability criteria which are false for sets are valid for currents. In the presence of suitable results on differentiability of Lipschitz mappings these classes of currents may be defined in a more customary way via exterior algebra and integer currents are those whose density is an integer multiple of the corresponding area factor. (The area factor is related to the Jacobians mentioned above

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as well as to the problem of finding optimal rectangles enclosing a given convex body, which was studied, for example, in [39].) The Plateau problem may be stated in the full generality of a complete metric space E: given a k-dimensional integer rectifiable current S with compact support and zero boundary, find a (k + 1)-dimensional integer rectifiable current T having the least total mass such that OT = S. In general, there may exist no currents T with OT = S, but such currents always exist if E is a Banach space. However, to assure the existence of a suitable minimizing sequence (whose limit would give a solution to the Plateau problem under fairly general assumptions, for example, if E is dual of a separable space) one needs that the following isoperimetric inequality holds: for every k-dimensional integer rectifiable current S with OS = 0 there is a (k + 1)-dimensional integer rectifiable current T with OT -- S and M ( T ) ~< c ( M ( S ) ) (k+l)/k, where c is a constant depending on E and k only. Whether this holds in every Banach space is an open problem; it has been proved in duals of separable Banach spaces having a weak* finite-dimensional decomposition.

2. Infinite-dimensional geometric measure theory The relation between infinite-dimensional measures and geometry of the infinite-dimensional Banach space E is much weaker than in the finite-dimensional case. Consider just the problem of describing the image of a non-degenerated Gaussian measure V on E by a continuous linear transformation T: even for simple transformations, such as T x -- 2x, the image is singular with respect to ?, and so no analogue of the classical substitution theorem can hold for such transformations. The same situation occurs for most shifts. The most common setting is therefore not only a Banach space E equipped with a measure lz, but also with a vector space H of the set of directions in which/~ behaves invariantly; it is also often assumed that H is (a continuous image of) a Hilbert space. (The basic example is, of course, a Gaussian measure y in E with H being its Cameron-Martin space.) The (possibly non-linear) transformations of the form x ~ x + h (x), where h : E ---> H are the natural candidates for which the substitution theorem may be valid. The role of geometry of E has nearly disappeared and in fact E is usually just assumed to be a locally convex space. Below we comment on the background of the basic concepts of derivative of measures in Banach spaces and briefly indicate some directions of research. Then we discuss results showing that not only covering theorems but even some of their natural corollaries often fail in infinite-dimensional situation even for Gaussian measures.

2.1. Differentiable measures In much of modern analysis in finite-dimensional spaces, the role of pointwise derivative has been completely overshadowed by that of derivative in the sense of distributions. If f : R --+ R is Lebesgue integrable, its distributional derivative may be defined as a Lebesgue integrable function g : R -+ ~ such that the formula for integration by parts

f

op' (t) f (t) dt = -

f

cp(t)g(t) dt

(1)

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holds for every smooth 4~ : I~ --+ I1~with bounded derivative. However, observing that in (1) the functions f and g are only used as acting on functions by integration, i.e., as measures, we may consider it as defining that the distributional derivative of a signed measure/1, is a signed measure v such that that f 4/(t) d/z(t) -- - f 4~(t)dr(t) for every smooth bounded 4~ :I~ --+ I~ with bounded derivative. On the real line, this generality is partly spurious, since it is easy to see that a measure p~ on I~ has this derivative if and only if it is a function of bounded variation. (Somewhat loosely, one says that the measure lz(E) = fE f(t)dt is a function, namely, the function f . ) However, the derivative may well be a measure which is not a function, for example, the derivative of the function f(t) = signum(t) is the Dirac measure. A similar approach is used in ]1~n t o define distributional partial derivatives; and again their existence means that the measure is a function. In fact, it is again a function of bounded variation, usually by definition (see, for example, [61 ]). The definition of distributional derivatives of measures admits a direct generalization to Banach spaces (where we have no notion of a measure being a function): the derivative of a (finite Borel) signed measure p, in direction w is a signed measure D,,,# such that

f D,,,qb(x)d # ( x )

= -

f

~ , x , dD,,,/z(x)

(2)

for every bounded continuously differentiable 4~: X -+ I1~ with bounded derivative. The definition immediately implies that the set of directions of differentiability of p, is a linear space, the mapping w--+ D,,,~t is linear and that differentiation commutes with convolution, i.e., D,,,(v 9 lz) = v 9 D,,,lz provided that D,,,/z exists. Directional derivatives of measures may be equivalently defined by more direct fbrmulae: derivative of/z in the direction w in Skorochod's sense is defined by (see [52, w ] for details)

f ckdD,,,# = -

lim r--~ O

f

+rw)-

ok(x) d/z(x)

(3)

r

provided that the limit exists for every bounded continuous 4~:X ~ 11~;the functional defined by the limit is necessarily an integral with respect to a measure. Another approach which was developed in finite-dimensional spaces by Tonneli needs essentially no modification in infinite-dimensional spaces: we require that p, has a disintegration

f qbdlz = fvfRqb(y + tw)~P~'(t)dtdv(y), where Y is a complement of span{w}, v is a probability measure on Y and ~ , are (right continuous) functions of bounded variation; under these conditions we define

f~dD,,,~:frf~(y+tw)d~(t)dv(y).

(4)

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It is easy to see that the derivatives of measures defined by (3) or (4) satisfy (2). If (2) holds, we obtain (3) by denoting vr(E) = ~.{t 9 [0, r]: tw 9 E}/r, inferring from the formula for differentiation of convolution that

f q~dvr * Dwlz = - f qb(x + rw) -qb(x) d/~(x) Y

first for every bounded continuously differentiable 4} : X --+ IR with bounded derivative and then, by approximation, for every bounded continuous q~:X --+ R, and by letting r ~ 0. Finally, to obtain (4) from (2), we disintegrate

fhdDwlz=fyfRh(y+tw)d{ry(t)dv(y) and let ~py(t)= O'y(-Cx~, t]. By approximation, it suffices to show that (4) holds for every continuously differentiable function 4}:X ~ IR with bounded derivative and with {t 9 R: ck(y + tw) ~ 0 for some y 9 Y} bounded. For any r > 0 denote gr(Y + tw) = ft__~ ck(y + sw) - r + (s + r ) w ) ds and use (2) and integration by parts to infer that

fD,,,gr(x)dlt(x)=-fgr(x)dD,,,rt(x)---frfRgr(y+tw)dcfx.(t)dv(y) =fvfRD,,,gr(y+tw)q/,'(t)dtdv(y) 9 Since D,,,gr(x) = r - 4 ' ( x + rw), (2) follows by letting r --+ cx~. Currently the most useful notion of derivative of a measure # (often called differentiability in the sense of Fomin) is obtained by requiring additionally that D,,,# be absolutely continuous with respect to #. This is equivalent to validity of (3) for every bounded Borel measurable function or to differentiability at t = 0 of the function assigning to t 9 I~ the measure # shifted by tw when the space of measures is equipped with the usual norm. The Radon-Nikod~m derivative of Du, # with respect to # is called the logarithmic derivative of # in direction w; one readily sees that this term is justified in the finite-dimensional situation. All these notions have been treated as a special case of differentiability of mappings of the real line into the space of signed measures equipped with various topologies in [50]; another particular case of this treatment is the notion of differentiability of measures along vector-fields. (Of course, in this generality some of the equivalences mentioned above may fail.) Under very mild assumptions, these authors also prove the key formula d/~a/d#h = exp(f~ O~(x)dt), where Ot is the logarithmic derivative of t 9 IK -+ ~t. (See [50] for the history of this formula and its applications.) In the setting when H is a subspace of E consisting only of directions of logarithmic differentiability and h : X -+ H, one can, under appropriate assumptions, compute the logarithmic derivative of t -+ (id + th):# and the Radon-Nikod~m derivative d(id + th):#/d# from the derivative of h and directional logarithmic derivatives of # - the latter gives a substitution theorem mentioned above (see [51]). The assumptions alluded to here are, of necessity, much stronger than those mentioned so far: since the formulas involve either the trace of the derivative of h in

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the direction of H or the determinant of id + th' (x). (This also explains why H is supposed to carry a Hilbert space structure.) A Gaussian measure is logarithmically differentiable exactly in the directions of its Cameron-Martin space and the derivatives may be found explicitly. For these measures, the above results form just a beginning of the story; see, for example, [8] for much more. The natural problem of unique determination of a measure by its logarithmic derivative has been answered negatively in [38]. (Prior to it, several authors noted that a positive answer would not only mean that some correspondence between functions and measures survives to the infinite-dimensional situation but would also have interesting applications.)

2.2. Surface measures Several approaches have been suggested to the definition of the surface measure induced by a given measure /z on E. A natural way is to assume that the surface is defined as {x: qg(x) = 0} where qg:E --+ ]1{ is such that for sufficiently many functions g on E the measures qg=(gtz) have continuous density k e with respect to the Lebesgue measure; the value of the surface integral of such g is then k e(O)/kl(O). To prove the assumption of continuity of k e, one may use differentiability of ~ together with the Malliavin method. (More details may be found in [8].) Uglanov's method [56] is based on the idea that, if/z has logarithmic derivative in direction w and G is the graph of a smooth function from a complement of IRw to Rw then D,,lz{a + tw: a E A, t <~ 0} should be a measure on G which is absolutely continuous with respect to the corresponding surface measure on G with known Radon-Nikod~m derivative. This can be used to define the surface measure of subsets of G provided that it does not depend on the choice of w. This independence is shown under suitable assumptions, which appear less stringent than in other methods. It is natural to imagine that the theory of surface measures (or, more generally, measures on surfaces of finite co-dimension), could be understood also as theory of integration of differential forms and/or currents of finite co-degree. Such possibility has been explored in a series of papers starting from [49].

2.3. Measures and balls Only little seems to be known about the interplay between geometry of an infinitedimensional Banach space X and behaviour of measures on it. A reasonably clear picture showing that the situation is rather complicated has been obtained concerning the questions that developed from the attempts to find valid infinite-dimensional counterparts to the differentiation theorem for measures according to which in finite-dimensional Banach spaces the Radon-Nikod3~m derivative g of a Borel measure # with respect to a Borel measure v is, at almost every x, obtained by the limit of the ratio of their averages on balls around x: g(x) = l i m r ~ 0 / z ( B ( x , r ) ) / v ( B ( x , r)). It turned out that the differentiation theorem for measures holds for all Borel measures in a Banach space if and only if it is finite-dimensional; a similar statement holds even for complete metric spaces (and for

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other families than balls) with a suitable combinatorial definition of finite dimensionality. (Like most results mentioned in this subsection, this may be found in [36].) Moreover, in a Hilbert space even much weaker statements are false as is shown by the following rather involved example from [43]. (Some numerical constants in the construction in [43] should not be taken too literally; a more accessible construction should eventually appear in [36].) There are a G a u s s i a n measure v in ~2 a n d a p o s i t i v e f u n c t i o n f ~ L i (v) such that fB(x,r) f d v / v ( B ( x , r)) tends to infinity as r tends to zero not only f o r every x but even uniformly on g.2, in other words /,

lim inf ]

f dv/v(B(x,

r)) -- oe.

(5)

r--+O x Ef 2 J B ( x , r )

A m o n g the possible corollaries of the differentiation theorem for measures that are not negated by the above example the most natural one is the determination of measures by balls" the differentiation theorem applied to # and # + v gives that # - 89 + v), hence # -- v whenever # and v coincide on all balls. The hope that at least this statement holds in general metric spaces was dashed by Davies in [ 14]" there is a c o m p a c t metric s p a c e on which two different Borel probability m e a s u r e s coincide on all balls. The basic idea of this beautiful example is the construction, for any given c~, fl > 0, of two measures # , v on a finite metric space M in which the only distances are one and two (so M is easy to imagine as the vertex set of a graph; points joined by an edge have distance one, remaining points have distance two) which coincide on all closed balls with radius one and satisfy 0 /~(M) -- c~ and v ( M ) -- [4. Such a space is obtained as a graph on n + n- vertices consisting of a complete graph on n 'inner' vertices, to each of which n different 'outer' vertices are joined. The main observation is that each ball of radius one consists either of one inner and one outer vertex, or of n inner and n outer vertices; then a straightforward calculation gives/z and v provided that n is large enough. (For example, all inner vertices may have # measure t~/2n and v measure t~/2n - ([3 - c~)/(n z - n), and all outer vertices may have # measure u / 2 n 2 and v measure u / 2 n 2 + ([3 - ct)/(n 2 - n).) Replacing recursively points by rescaled copies of such spaces, one finds a compact metric space M0 of diameter one on which two different Borel measures/x0, v0 coincide on all balls of radius less than one. The final space and measures are obtained as M0 U M i , / z 0 + vl and v0 + # l , where Mi is another copy of M0 (with corresponding measures/z I, vl ) and the distance between points of M0 and Mi is defined as one. The above construction showing that even the determination of measures by balls may be false in general metric spaces clearly leads only to highly non-homogeneous spaces and it cannot produce, for example, a Banach space. Indeed, for Banach spaces such an example does not exist ([45])" two finite Borel m e a s u r e s # a n d v coinciding on all balls in a separable B a n a c h space X necessarily coincide on all Borel subsets o f X. The argument blows suitably placed balls to show that # and v coincide on many convex cones that factor through a finite-dimensional subspace; a simple consideration of finite-dimensional projections o f / z and v then gives that they coincide on all half-spaces given by linear functionals belonging to a weak* dense subset of the dual unit ball. Hence # and v have the same Fourier transform, and the statement follows.

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It is an open problem if the determination of measures by balls holds under relatively weak homogeneity assumptions, e.g., in complete separable metric Abelian groups. In particular, except for a few special cases it is not known whether two finite Borel measures in a Banach space coincide provided that they coincide on all balls with radius at most one. (Note that by the above proof this holds for balls of radius at least one.) This motivates the attempts to prove the Banach space result without the use of Fourier transform. One such attempt noted that it would be enough to show that the family of Borel sets in a separable Banach space X is the smallest family of subsets of X containing balls and closed under complements and countable disjoint unions; the latter family is necessarily closed also under countable monotone unions and intersections. It has been recently shown that this statement holds in finite-dimensional spaces (it is not easy; both existing proofs ([20] and [60]) use Besicovitch's covering theorem), but not in an infinite-dimensional Hilbert space (any non-trivial intersection of two balls supplies a counterexample [25]). The extent of the difference between finite- and infinite-dimensional situation is apparent from the following two amusing statements concerning separable Hilbert spaces H: (A) The statement"whenever/~(B) ~> v(B) for all balls B with radius ~< 1, then # ~> v" holds if and only if H is finite-dimensional. (B) The statement"whenever/~(B) ~> v(B) for all balls B with radius ~> 1, then/z ~> v" holds if and only if H is infinite-dimensional. The statements considered in (A) and (B) are sometimes called positivity principles for small and large balls, respectively. The positivity principle for small balls follows from the differentiation theorem for measures, and so it holds in all finite-dimensional Banach spaces. The example behind the other implication of (A) uses the Gaussian measure v in e2 and positive function f 6 L i (v) such that (5) holds: a simple modification achieves f f dv < 1 and futx.r) f dv > v(B(x, r)) for r <~ 1" and the example needed for (A) is obtained with/~(A) -- fa f dr. This example also answers negatively the question of validity of positivity principle for all balls in general Banach spaces: In the space ~2 ~ II~ consider vl + v-i + #0 and v0, where the index r indicates the image measure under the mapping xEg~2----~xOr.

The statement (B) is much easier: in the n-dimensional case one may consider for ~ the Lebesgue measure on a ball and for v a small multiple of the Dirac measure in its centre. In the infinite-dimensional case one shows that the characteristic function of each cylinder (set of the form Jr -I (B), where 7r is an orthogonal projection with a finite-dimensional range and B is any ball in the range), can be obtained as a limit of functions from the convex cone generated by characteristic functions of balls with radius ~> 1. This reduces the proof to the finite-dimensional case of small balls. The negative results mentioned above are remarkably unstable. Riss [48] shows that every Banach space may be renormed so that the positivity principle holds for large balls. Only future investigations may reveal the fate of this little corner of geometric measure theory. It is possible that useful connections exist to probability theory on Banach spaces, especially to problems of large deviations; a small indication of this may be given by the use of Chernoff's theorem in [15] to prove the determination of measures by balls for measures with finite Laplace transform, or by the fact that more information may be sometimes obtained by using large balls instead of small balls.

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2.4. Differentiation theorems f o r Gaussian measures The importance of Gaussian measures justifies their separate treatment, although from the point of view of the problems considered here the results are rather sparse. It is not difficult to see that Gaussian measures on Banach spaces are determined by their values on balls of radii at most one [9]. Nevertheless, in 2.3 we noted that even for a Gaussian measure v in a separable Hilbert space the differentiation theorem for L I functions may fail not only almost everywhere, but even uniformly (see Eq. (5)). The density theorem may fail as well [42]: there are a Gaussian measure v in ~2 and a Borel set E C ~2 of positive v measure such that l i m r ~ 0 v ( E • B(x, r ) ) / v ( B ( x , r)) = 0 for v almost every x, although here it is not known if the failure may be uniform. On the other hand, by [55] there are infinite-dimensional Gaussian measures in separable Hilbert spaces f o r which the differentiation theorem holds for all functions from L p for p > 1; a sufficient condition is that the eigenvalues ok of their covariance satisfy ~j +l <~ccrj/j~ for some ~ > 5/4. The proof is rather technical and its important ingredient is the dimension-independent estimate of the Hardy-Littlewood maximal operator from [54].

3. Exceptional sets Here we discuss several notions of "null" or "negligible" sets in a Banach space that appear as exceptional sets in various questions of behaviour of mappings between infinitedimensional spaces. With the notable exception of the topological notion of the sets of the first catego~., these involve metric or linear conditions and sometimes behave in an unexpected way. First category sets are, unfortunately, rarely useful in problems where main point is to capture some of the roles played in finite-dimensional spaces by the Lebesgue null sets. When convenient, we will give the definition for Borel sets only with the understanding that a possibly non-Borel set is null if it is contained in a Borel null set. (However, it should be pointed out that Borel measurability is not a point of pedantry; leaving it out may easily not only change the definitions but render them meaningless.) The most appealing replacement for Lebesgue null sets in infinite-dimensional Banach spaces or even in complete separable metric Abelian groups is due to Christensen [ 12]. Let G be an Abelian topological group whose topology is metrizable by a complete separable metric. A Borel set E C G is Haar null if there is a Borel probability measure # on G such that every translate of G has measure zero. (Sometimes these sets are referred to as Christensen null. They have also been rediscovered under the name of shy sets.) Haar null sets form a o'-ideal since, if E,, are Haar null and /x,, are the corresponding measures, the measure obtained as an infinite convolution of suitable portions of # , witnesses that [,_J~ En is Haar null. If the group is locally compact, these null sets coincide with those H= I of Haar measure zero. If, however, the group is not locally compact, then every compact set is Haar null; this follows from the following important generalization of Steinhaus's theorem: if a Borel set A is not Haar null then A - A contains a neighbourhood of the identity in G. Haar null sets have been, deservedly, investigated in their own right. Since measures on G are inner regular with respect to compact sets, any subset of G containing a translate of

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every compact set is not Haar null. This has been used in [53] to show that if G is not locally compact, then it contains an uncountable collection of disjoint closed subsets which are not Haar null. Reflexive Banach spaces have been characterized as those in which every closed convex set with empty interior is Haar null, see [35] and [34]. Relatively simple examples show that a Fubini-type result is false for Haar null sets. A property useful in application to differentiability is that a Haar null set is Lebesgue null on lines parallel and arbitrarily close to any given line. Although the knowledge that an exceptional set is Haar null suffices for many applications (in particular, if we just need to have one non-exceptional point), there are situations in which this is not the case. Several seemingly different stronger notions of negligible sets in a Banach space X were defined (implicitly or explicitly) to improve, in particular, the resuits on Gfiteaux differentiability of Lipschitz functions. Recall that a cube measure in X is any image of the product Lebesgue measure on [0, 1]• under the mapping t ~ x + ~ tkxk provided that y~ Ilxk II < ~ . A cube or Gaussian measure is non-degenerate if every closed hyperplane gets measure zero. A Borel set E is a separable Banach space X is called (a) cube null if it is null for every non-degenerate cube measure on X [32], (b) Gauss null if it is null for every non-degenerate Gaussian measure on X [40], and (c) Aronszajn null if for every sequence u, ~ X with dense span, E can be written as a union of Borel sets E,, such that the intersection of E,, with any line in the direction u,, is of one-dimensional Lebesgue measure zero [3]. Clearly, Aronszajn null sets are Gauss as well as cube null. It is in fact not difficult to see that Gauss and cube null sets coincide, and may be equivalently defined as those Borel sets that are null for every measure with a dense set of directions of differentiability (cf. 16]). A remarkable result of Cs6rnyei [13] shows that every cube null set is Aronszajn null, and so all these notions coincide. The main difficulty of the proof stems from the requirement that the sets E,, be Borel; note however that without assuming this the definition (c) would become meaningless since by an observation from 161 the whole space would be Aronszajn null. Clearly, (a) or (b) immediately show that every Aronszajn null set is Haar null. The converse is false because there are compact Aronszajn non-null sets (e.g., cubes) while every compact set K is Haar null. This example also shows that the analogue of Steinhaus's theorem is no longer valid for Aronszajn null sets. As one would expect, in finite-dimensional spaces Aronszajn null sets coincide with Lebesgue null sets. By an example from [7], Aronszajn null sets are not invariant under C ~ Lipschitz isomorphisms. It tbllows that these sets cannot provide lull characterization of exceptional sets that are invariant under such mappings (such as sets of non-differentiability). A possible remedy, suggested by Bogachev [7], is to consider sets which are null for all measures differentiable with respect to a spanning sequence of vector-fields u,, (i.e., such that for every x the vectors u,,(x) span X). Another possible remedy, from [47], is to consider sets which can be written as a union of Borel sets E,, such that, for some u,, 6 X and e,, > 0, the intersection of E,, with any curve y :IK ~ X with Lip(t ~ y ( t ) - tu,,) < e,, is of onedimensional measure zero. Many questions concerning these sets are open. For example, it is not known if a formally stronger notion in the spirit of (c) is equivalent to the one described here or if an analogue of Cs6rnyei's result holds in this situation. In fact, it is

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not even known if in finite-dimensional spaces (of dimension at least three) these null sets coincide with Lebesgue null sets. Intriguing invariance problems for the null sets defined above remain open and deserve to be mentioned because of their possible application to the problem of classifying Banach spaces up to Lipschitz isomorphisms. We have already noted that Aronszajn null sets are not invariant under Lipschitz isomorphisms. The same holds for Haar null sets; moreover, in [28] there is an example of a Haar non-null set in ~2 which can be transformed by a Lipschitz isomorphism into a hyperplane; note that hyperplanes are not only Aronszajn null, but have to be null for any notion for which we wish to have any reasonable statement on differentiability of Lipschitz functions almost everywhere. For the intended application to the Lipschitz isomorphism problem it would be enough to know that a null set cannot be transformed into a set whose complement is null; as the title of [28] indicates, even this may happen for Haar null sets, but it is unknown for Aronszajn null sets. No pertinent examples are known for the non-linear notions of null sets. We now briefly describe another appealing notion of exceptional sets which is a metric strengthening of the notion of first category sets; unfortunately, for our purposes, it has the disadvantage that it cannot describe non-differentiability sets of Lipschitz functions since these may well be of second category. These sets, however, play an important role in studying more exceptional behaviour (for example, non-differentiability of continuous convex functions). Out of the huge number of non-equivalent notions (cf. [58]), the two most natural ones (in our context) are or-porous sets and cr-directionally p o r o u s sets. It suffices to define the notions of porous and directionally porous sets only, since the prefix 'or' means 'the union of countably many of.' A set E C X is p o r o u s if there is 0 < Z < 1 such that for every x 6 E there are x,, E X converging to x and r,, > Z IIx,, - x II such that the balls B ( x , , r,,) are disjoint from E. It is porous in direction u if the x, may be found on the line through x in direction u, and it is directionally p o r o u s if there is u such that it is porous in direction u. The same notion of a-porosity (though a different notion of porosity) is obtained if we define porosity with Z independent of x; for a-directional porosity we could even allow dependence of u on x. The porosity a-ideals are Borel: every cr-(directionally) porous sets is contained in a Borel cr-(directionally) porous set. Clearly, cr-directionally porous sets are a-porous, and a-porous sets are first category. Every cr-directionally porous set is Aronszajn null: if E is porous in direction u, it is porous in every direction sufficiently close to u; so, if/z has a dense set of directions of differentiability, then E is porous in a direction v of differentiability of ~z and the disintegration along this direction shows that # ( E ) = 0. In finite-dimensional spaces the notions of a - p o r o u s and cr-directionally porous sets coincide, otherwise they differ: by [46], every infinite-dimensional space is a union of a a-porous set and of an Aronszajn null set. For super-reflexive spaces a much stronger decomposition statement (using sets that are sometimes called strongly very porous) can be found in [33]. (Both these statements have been used to give examples concerning nondifferentiability, and so we will meet them again.) These constructions are based on the trivial observation that a Borel set which meets every k-dimensional affine subspace (where k is fixed) in a Lebesgue null set is necessarily Aronszajn null. Our next classes of exceptional sets consist of very small sets indeed (at least compared to the previous classes). Their importance stems from the fact that they have been used in the only results in differentiability in which we have complete characterizations (see

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Section 4.1.1). They are the a-ideals of sets that can be covered by countably many k codimensional Lipschitz, respectively g-convex, hypersurfaces. The k codimensional Lipschitz, respectively g-convex, hypersurfaces are defined as sets of the form {w + 4~(w): w 6 W}, where X is a direct sum X = W ~ U and 4~: W w-~ U is Lipschitz, respectively g-convex. Recall that a Lipschitz mapping 4~ : X ~ Y is g-convex if for every y* 6 Y* the composition y* o 4~ may be expressed as a difference of two continuous convex functions. These ~r-ideals are Borel and are properly contained in all the previous a-ideals; they are becoming smaller with k increasing, and those defined using g-convex hypersurfaces are properly contained in the ~r-ideals defined via Lipschitz hypersurfaces. In case k = dim(X), both notions give just countable sets. The prefix 'k-codimensional' will be omitted if k = 1. Finally, we meet a cr-ideal combining measure and category in a useful (and non-trivial way). It has been used in [31] to obtain first (and so far only) infinite-dimensional results on Frgchet differentiability of Lipschitz functions almost everywhere. The basic idea is to consider a suitable completely metrizable space of measures differentiable in direction of a spanning sequence of vector-fields (which may depend on the measure) and define that a set is null if it is null for residually many of these measures. This, however, appears to be technically complicated, and so the definition uses a parametric approach (in which the condition of differentiability of measures is not so apparent). Let r = [0, 1]r~ be endowed with the product topology and the product Lebesgue measure #, and let F ( X ) denote the space of continuous maps y : r ~ X having continuous partial derivatives. We equip F ( X ) with the topology of uniform convergence of the maps and their partial derivatives and define a Borel set E C X to be F-null if

r,{, z.

E}-0

for residually many y 6 F (X). Note that, since F (X) is completely metrizable, the family of null sets forms a proper a-ideal of subsets of X. A simpler but less useful variant of the notion of F-null sets may be obtained by replacing X' by [0, 1]k. These notions of F-null sets can be viewed as a special case of a general scheme defining negligibility of a set A in the space by requiring that it is negligible inside all except negligibly many elements of the hyperspace (space of subsets, space of measures). For example, the set 72(X) of Borel probability measures on a complete separable metric space X considered as a subset of the dual to the space of bounded continuous functions with the weak* topology is completely metrizable, so one can try to consider as negligible those sets A C X that satisfy ~(A) = 0 for residually many ~ E 7:'(X). Another, purely topological, example can be obtained by using the space/C(X) of non-empty compact subsets of X equipped with the Hausdorff metric and defining negligibility of a set A C X by requiring that C t~ A is of the first category in C for residually many C 6/C(X). The usefulness of the notions introduced in these two examples is somewhat diminished by the easily seen tact that for Borel sets they are both equivalent to the notion of the sets of the first category. (In this connection, one may note that, without any condition on a set A C X, A is of the first category if and only if C N A = 0 for residually many C 6/C(X). This follows immediately by noting that the union of the compacts belonging to a G~ subset of/C(X) is a Suslin set.) Nevertheless, the negligibility notion from the second example (in possibly non-separable spaces) and its variants have been successfully used in [37] to

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show existence of points of G~teaux differentiability of continuous convex functions on certain non-separable Banach spaces (where the set of points of differentiability need not be Borel).

4. Differentiability We will consider the question of existence of points of Fr6chet and G~teaux differentiability of continuous convex functions and of Lipschitz mappings between Banach spaces. For definitions, basic information and, in particular, for the concept of Radon-Nikod3~m property of a Banach space X and the result that all Lipschitz mappings of finite-dimensional spaces into X are differentiable (at least at one point or, equivalently, almost everywhere) see Section 7 of [21 ].

4.1. Convex functions The study of differentiability problems for continuous convex functions is greatly simplified by several facts (cf. [21]): the sets of points of G~teaux as well as of Fr6chet differentiability are G~ (the latter even in non-separable spaces), if one term of a sum of such functions is (G~teaux or Fr6chet) non-differentiable at x, then the sum is non-differentiable at x, at every point the one-sided directional derivatives exist and form a convex continuous and positively l-homogeneous function of the directions, hence G~teaux differentiability at x is equivalent to the requirement that f ( x + th) + f ( x - th) - 2 f ( x ) = o(t) as t --+ 0 and Fr6chet differentiability is equivalent to the requirement that f ( x + h) + f ( x - h) - 2 f ( x ) = o(llhll) as h ~ 0, the subdifferential O f ( x ) --- {x* E X*: x * ( u ) <~ f ( x + u) - f ( x ) for all u 6 X} is non-empty and differentiability has a simple description as a property of the subdifferential: f is G~teaux differentiable at x if and only if its subdifferential at x is a singleton and f is Fr6chet differentiable at x if and only if the multi-valued mapping y --+ O f ( y ) is single valued and norm continuous at x, i.e., for every e > 0 there is ~ > 0 such that O f ( y ) C B(x*, e) for all y E B ( x , 8) and x* ~ O f ( x ) . A number of results on convex functions has been generalized to statements about monotone operators: a mapping T of a set E C X to the family of non-empty subsets of X* is called monotone if (y* - x * ) ( y - x) >~ 0 for all x, y E E, x* ~ T ( x ) , and y* ~ T ( y ) . (Note that some authors require E = X but allow T ( x ) to be empty.) Basic properties and references to situations in which they play a significant role may be found in [41.]. Important examples of monotone operators are provided by subdifferentials of continuous convex functions. Standard results on convex functions have their counterpart in the theory of monotone operators. For example, the simple but useful fact that continuous convex functions are locally Lipschitz may be obtained as a corollary of the fact that monotone operators on open sets are locally bounded. Another direction in which the subdifferential approach may be understood is via selection theorems. It is easy to see that the mapping T(x) = O f ( x ) is weak* upper semicontinuous (i.e., the set {x: T(x) C G} are open for every weak* open G C X*) and has non-empty weak* compact values (we abbreviate both these properties of T by saying

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that T is weak* usco). If ~ : X --+ X* is a selection for T (i.e., ~(x) E T ( x ) for all x) which is norm-to-weak* continuous at x, then f is G~teaux differentiable at x; if it is even norm-to-norm continuous, then f is Fr6chet differentiable at x. Since the T given by the subdifferential is locally bounded, one may obtain differentiability results from purely topological statements on the existence of selections of weak* usco mappings of, say, topological spaces having the Baire property into the unit ball of X*. This approach gives also results for monotone operators, since maximal monotone operators an open sets are (locally bounded and) weak* usco. The price paid for the higher generality of the approach is weaker information about the size of the set of points of differentiability; in the selection approach one may hardly expect stronger exceptional sets than those of the first category. 4.1.1. G~teaux differentiability o f convex functions The remarkable results of [59] give a complete description of the size of sets of points of G~teaux non-differentiability of continuous convex functions. They considerably strengthen a series of previous infinitedimensional results starting with Mazur as well as more detailed previous results in the finite-dimensional case. The set o f points o f G~teaux non-differentiability of an arbitrary continuous convex function on a separable Banach space X can be covered by countably many 6-convex hypersurfaces. Conversely, for every set E contained in countably many 6-convex hypersurfaces there is a continuous convex function on X which is G6teaux nondifferentiable at every point qf E. Note that in case dim(X) -- 1 we recover the classical statement that the sets of points of non-differentiability of convex functions on ~ are exactly countable sets. The natural generalization of the question answered in the previous paragraph is the study of those points at which the subdifferential is large. Again, [59] gives a complete answer, which reduces to the previous statement if k = 1: f o r an arbitral, continuous conw, x function on a separable Banach space the set of those x at which the dimension of the affine span of the subd~fferential is at least k can be covered by countably many k-codimensional 6-convex hypersurfaces. Conversely, f o r every set E contained in countably many k-codimensional 6-conw, x hypersurfaces there is a continuous convex function on X such that f o r every x E E the dimension of the affine span o f Of(x) is at least k. To indicate the way in which this is proved, let N denote the set of points x for which the dimension of the affine span of Of (x) is at least k. Given a k-dimensional subspace U of X, u* E U*, and ~ > 0, the set B of those x E X for which there is x* E 3 f ( x ) extending u* such that IIx*l] < l / e and f ( x + h) - f ( x ) >~x*(h) + ellhll for all h E U is a subset of N; moreover, by separability, N is a countable union of such sets B. It is therefore enough to consider one such set B and show that it is covered by a k-codimensional 6-convex hypersurface (which still needs work). To show the converse it suffices to consider the case of one hypersurface; then one can define the required convex function using the convex functions describing the hypersurface. The above discussion of Gfiteaux differentiability may be modified to show that f o r any monotone operator T on an open subset of a separable Banach space the set of points at which T ( x ) contains at least k affinely independent elements can be covered by countably many k-codimensional Lipschitz hypersurfaces. Applying this fact to the subdifferential of a convex function gives, however, only a weaker version of the above results.

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Let us briefly mention some points of the non-separable theory; for more information see [16,18] or [41]. X is said to be a weak Asplund space if every continuous convex function on X is G~teaux differentiable on a residual set and a G~teaux differentiability space, if every continuous convex function on X is GS.teaux differentiable at least at one point (or, equivalently, on a dense set). An interesting class of weak Asplund spaces which includes all separable spaces as well as spaces admitting a Gfiteaux smooth norm (or just a GS.teaux differentiable Lipschitz bump function) is formed by those Banach spaces whose dual unit ball B* satisfies the condition (of nature similar to topological descriptions of Radon-Nikod3)m property of duals as in 4.1.2(e)) that in the following 'fragmentability' game the second player has a strategy guaranteeing that n~_-I Fk n G k is a singleton" the first player starts by choosing a non-empty weak* closed subset Fi of B*, then the second player chooses a weak* open set G l such that F! n G i ~ 0, then the first player chooses a weak* closed subset F2 of F! such that F2 N G l r 0, then the second player chooses a weak* open set G2 such that F2 N G2 5~ 0, etc. This implies that B* belongs to the so called Stegall's class, i.e., has the property that any weak* usco mapping of a Hausdorff topological space Z to B* has a selection which is weak* continuous on a residual set. With our definition, it can be shown by considering a minimal usco mapping of Z to B* and showing that it is single-valued on a residual set with the help of the B a n a c h - M a z u r game in Z. (No use of the fact that B* is a ball has been made; the argument works in any compact Hausdorff space satisfying the fragmentability condition.) Recent papers of Kalenda [23,24] and of Kenderov [26] show that these classes are different and do not coincide with weak Asplund spaces. Very recently Moors and Somasundaram [37] used the hyperspace based notions of negligibility mentioned at the end of Section 3 to answer the key open problem of the theory by producing a Gfiteaux differentiability space that is not weak Asplund. 4.1.2. Fr~chet differentiability o f convex functions We have seen in [21] which separable spaces have the property that every continuous convex function on them is Fr6chet differentiable on a dense G~ set; they are precisely those whose dual is separable. Other characterizations follow from this, and similarly satisfactory results hold in non-separable setting. Banach spaces in which continuous convex functions have points of Fr6chet differentiability are called Asplund spaces. They are characterized by any of the following equivalent properties: (a) Every continuous convex function on X has a point o f Fr~chet differentiability. (b) Every continuous convex function on X is Fr~chet differentiable on a dense G~ set. (c) The dual o f every separable subspace o f X is separable. (d) X* has the RNP. (e) For every non-empty bounded set E C X* and every ~ > 0 there is a weak* open set S meeting E such that S n E has diameter less than e. (f) For every non-empty weak* compact convex E C X* and every e > 0 there are u ~ X and 6 > 0 such that the weak* slice S ( E , u, 6) = {x* ~ E: x*(u) >~ sup~,.~E y*(u) - 6} has diameter less than e. The equivalence of (a) and (b) follows from the set of points of Fr6chet differentiability being G~; see [21 ]. From the negation of one of the statements (c)-(e) one may prove the negation of (f); if then E is a non-empty weak* compact convex set without small weak* slices, the function f (x) = suPx. ~ E x* (x) is nowhere Fr6chet differentiable, and so is the sum of the original norm with f (x) and f ( - x ) . In this way, we even see that every non-Asplund space admits an equivalent norm satisfying lim suPh_,0(llx + hll + Ilx - hll - 21lxll)/llhll > e for

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some e > 0; such norms are sometimes called e-rough. To prove, say, (e) =~ (b), one may use (e) to deduce that every weak* usco mapping of a Hausdorff space Z to B* has a selection norm continuous on a residual set; this is a purely topological statement whose proof and use is similar to what was described in 4.1.1 for G~teaux differentiability. Another possibility is to prove the statement in the separable case only (see later), and use the method of separable reduction. (For a general approach to this method see [10].) The question of describing the size of the set of points of Fr6chet differentiability of continuous convex functions is not yet fully understood even in the separable case. However, we have: the set of points of Fr~chet non-differentiability of any continuous convex functions on a separable Asplund space is a-porous. To see this, let x k* be dense in the dual and let Fk,/ be the set of those x in whose neighbourhood the Lipschitz constant of f does not exceed 1 and for which there is x* ~ Of(x) such that IIx~ - x * 11 < 1 / l and there are arbitrarily small h such that f ( x + h) - f ( x ) - x*(x) > 4[[hll/l. Then for any y ~ B(x + h, [[h[[/l 2) and any y* ~ Of(y) we have y*(h) >t f (y) - f (y - h) >~ f (x + h) - f (x) - 2[[h[[/l >~x~.(h) + [[h[[/l; so [[x[ - y*]] ~> 1/l and y r Fk,/. Hence B(x + h, [[h[[/l 2) A Fk,/ = 0, which shows that Fk,/ is porous. The proof is finished by observing that the set of points of Fr6chet non-differentiability is covered by the union of Fk,/. The above argument may be modified to show that for every monotone operator on a Banach space with a separable dual there is a porous set outside of which the operator is single-valued and norm-to-norm upper semi-continuous. In both these results the porosity may be strengthened (to so-called o'-cone porosity, in which the holes, instead of balls, may be cones); for details see [41]. Although these notions are still some way from a description of the size of sets of Frdchet non-differentiability of convex functions, there is a (very strong) porosity condition which enables the construction of badly differentiable functions: if X is uniformly convex, r,, "N 0 and E C X is such that for every x 6 E and every s > 1 there are z,, such that [Iz,, - x l[ < ~r,~ and B(z,~, r,,/~.) N E = 0, then there is a continuous convex function on X which is Fr6chet non-differentiable at every point of E. The basic idea in construction of such a function is (assuming that E C B(0, 1)) to consider the supremum of all affine function majorized by the restriction of a uniformly convex function to E U (X \ B(0, 2)). A similar construction (together with adding the functions constructed for a sequence of such sets E) was used by Matou~kovzi [33] to show that on

every super-reflexive separable space there is an equivalent norm whose set of points of Fr~chet differentiability is Aronszajn null. This example, in particular, shows that the results mentioned so far are not strong enough to produce, given a convex continuous function g on a separable Asplund space X and a Lipschitz mapping f of X to an RNP space Y, one point x E X at which g would be Fr6chet differentiable and f G~teaux differentiable. This question was answered by [31 ]: the set ofpoints ofFr~chet non-differentiability ofany convex continuous function on a separable Asplund space is F-null. (See 4.2.1 for results on Gfiteaux differentiability implicitly alluded to here, and 4.2.2 for a more general version of this statement.) Note that the incompatibility of F-null sets and a - p o r o u s sets (which probably carries over to cr-cone porous sets) makes it is difficult to conjecture how a characterization of sets of Fr6chet non-differentiability may look like.

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4.2. Lipschitz functions One of the problems guiding the development of differentiability results for Lipschitz mappings is the Lipschitz isomorphism problem: if X, Y are Lipschitz isomorphic (i.e., if there is a bijection f ' X ~ Y such that both f and f - 1 are Lipschitz) are they linearly isomorphic? Much of the theory described below can be, and has been, successfully used to obtain partial answers. The basic idea is that the derivative of f should provide such an isomorphism. For this to work, one has to assume additional properties of X, Y such as the RNP (or just reflexivity, or even just super-reflexivity), since otherwise f may be nowhere differentiable. However, at the present time any direct use of this program comes to the obstacle caused by the open problem whether the G~teaux derivative of a Lipschitz isomorphism of ~2 onto itself is surjective at least at one point. Another approach to the Lipschitz isomorphism problem is based on the observation that a G~teaux derivative of a Lipschitz isomorphism f of X to Y is surjective provided that all compositions y* o f are Fr6chet differentiable for all y* e Y*. In fact, a dense set of y* suffices, which gives a good reason why we care so much about the problem of finding a common point of G~teaux differentiability of a Lipschitz mapping and of Fr6chet differentiability of countably many real-valued Lipschitz functions. In these arguments Fr6chet derivative may be replaced by almost Fr6chet derivative (see 4.2.4). The weakening of the concept of Fr6chet derivative can be pushed even further, to the so called affine approximation property of [4]; we will not consider these results here, but mention that in the super-reflexive case most applications of almost Fr6chet differentiability (see 4.2.4) to the Lipschitz isomorphism problem may be also obtained with the help of the uniform version of this property. We should recall that differentiability results for general Lipschitz functions cannot be obtained using sets of the first category, as there are Lipschitz functions f : R ~ R which are differentiable only on first category sets. In fact, on the real line these results cannot be obtained by any means weaker than the Lebesgue measure, since for every set N C I~ of Lebesgue measure zero there is a Lipschitz f : I~ --+ R which in non-differentiable at all points of N; moreover, by [57] the sets of non-differentiability of Lipschitz functions I~ --+ I~ are characterized as G ~ sets of Lebesgue measure zero. 4.2.1. Gfiteaux differentiability of Lipschitzfunctions The question how small are the sets of points of G~teaux non-differentiability of Lipschitz functions does not have a complete answer yet; it is not even known if in R" (n ~> 3) the o'-ideal generated by the sets of non-differentiability of real-valued Lipschitz functions coincides with the Lebesgue null sets. Nevertheless, the fact that locally Lipschitz mappings of separable spaces into RNP spaces are G~teaux differentiable outside Haar null sets mentioned in [21 ] or the following stronger result are sufficient for a number of purposes: every locally Lipschitz mapping of a separable space into an RNP space is Gfiteaux differentiable outside an Aronszajn null set. There are several remarkably simple proofs of this statement: the basic idea is that, if f :X --+ Y and Un e X have a dense span, the sets En of those x e X for which the directional derivative f ' ( x , Un) = l i m t ~ o ( f (x + tun) - f ( x ) ) / t does not exist are Borel and, by the RNP of Y, the intersection of En with any line in the direction Un is of onedimensional Lebesgue measure zero. It remains to show that the set of points of G~teaux

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non-differentiability at which directional derivatives exist in a spanning set of directions is Aronszajn null; this set is reasonably well understood, since: if f is a locally Lipschitz mapping o f a separable space X to Y, then the set o f those x ~ X at which f has the directional derivative in a spanning set o f directions but is not G~teaux differentiable is cr-directionally porous. To see this, one shows, for every u, v ~ X, y, z ~ Y and e, 6 > 0, the directional porosity of the set E of those xo 6 X such that Lip(f, B(xo, 6)) < 1/e, Ilf(xo + tu) - f (xo) - tyll + IIf(xo + tv) - f (xo) - tzll <<.eltl for Itl < 6 and there are arbitrarily small Isl such that IIf(xo + s ( u + v)) - f ( x o ) - s ( y + z)ll > 4elsl: if Isl is small and IIx - (xo + su)ll < eZlsl, then IIf(x + sv) - f ( x ) - szll >~ IIf(xo + s(u + v)) f (xo) -- s ( y + z)ll -- Ilf(x + so) -- f (xo + s(u + v))ll -- IIf(x0 + su) -- f (xo) -- syll -Ilf(x0 4- su) -- f(x)ll > Elsl, so x r E. The set in question is covered by countably many

such sets E since it suffices to consider u, v from a dense countable subset of X, y, z from a dense countable subset of the span of f (X) and rational e, 6. Intriguing questions are obtained when one attempts to use these results to answer the Lipschitz isomorphism problem. The G~teaux derivative of a Lipschitz isomorphism f of a separable Banach space X onto an RNP space Y, whenever it exists, is a linear isomorphism onto a closed subspace of Y (so X has the RNP as well). This subspace is complemented if, e.g., Y is reflexive (see [5]). Nevertheless, the following key problem is still open: if f is a Lipschitz isomorphism o f ~2 onto itself is there a point at which its G~teaux derivative is a linear isomorphism o f s onto itself? One may hope that the Gfiteaux derivative of any Lipschitz isomorphism between RNP spaces is, at least at one point, a linear isomorphism between them; this more general version of the problem is open as well. Since it is easy to see that if a Lipschitz isomorphism f of X onto Y is Gfiteaux differentiable at x and f - I at f ( x ) , then D r ( x ) is a linear isomorphism of X onto Y, a positive answer would be obtained if the null sets with respect to which one has the differentiability theorem were invariant under Lipschitz isomorphisms. We have, however, pointed out in Section 3 that this is not the case for Haar null nor for Aronszajn null sets. Other problems on invariance of null sets under Lipschitz isomorphisms treated in Section 3 have been also motivated by the Lipschitz isomorphism problem. For any given notion of null sets, the worst examples would be of the situation when a complement of a null set is mapped onto a set of G~teaux non-differentiability of some Lipschitz function; such examples are not known even for Haar null sets (and so also not for Aronszajn null sets). Curiously enough, if the Lipschitz isomorphism f : X ~ Y has all one-sided directional derivatives, then the image of the set at which g: X --~ Z (with RNP Z) is not Gfiteaux differentiable is even Aronszajn null: the image of the set of points at which g is non-differentiable at some direction is contained in the set of non-differentiability of g o f - I , and the remaining part of the set of non-differentiability points of g is a-directionally porous, so its f image is also a-directionally porous, since Lipschitz isomorphisms having one-sided directional derivatives map a-directionally porous sets to a-directionally porous sets. The Lipschitz isomorphism problem may well require a strengthening of the above results on Gfiteaux differentiability. This motivates the quest for finding smaller a-ideals of sets for which the differentiability statement still holds (and is genuinely stronger than the use of Aronszajn null sets). The non-linear concepts of Aronszajn null sets briefly discussed in Section 3 provide such a-ideals. From these results (or directly) is is also easy to see that every Lipschitz f u n c t i o n f f r o m a separable B a n a c h space X to an R N P space

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Y is GCtteaux differentiable F-almost everywhere. Note again that for these a-ideals the problem of invariance under Lipschitz isomorphisms is open. 4.2.2. Fr~chet differentiability of Lipschitz functions Until recently, the only general result on Fr6chet differentiability of Lipschitz mappings, except the case of finitedimensional domain where the concepts of G~teaux and Fr6chet differentiability coincide, was that every Lipschitz mapping f of an Asplund space X to R has points of Fr~chet differentiability; a small generalization (for X separable) assumes only the weak* closure of the set of of G~teaux derivatives of f norm separable. It is immediate to deduce from this that the set of points of Fr6chet differentiability must be uncountable in every non-empty open set, and stronger information on the size of this set can be obtained by use of the mean value estimate (see 4.2.3). The original proof of the Fr6chet differentiability result is rather involved [44]; a simpler (but not simple) proof from [30] is based on the following ideas (we assume X separable): denote by D f ( E ) the set of all Gfiteaux derivatives attained at points of E. Let El be a ball of radius one, Wi = D r ( E l ) and let u l 6 X be such that the slice S(WI, u l, 61) has a small diameter. One can show that there is 01 > 0 such that whenever D r ( x ) ~ S(WI, u i, OI), then lim suph~0 I f ( x + h) - f ( x ) - D f(x)(h)l/llhll is small. Then one defines E2 as the set of those x from the intersection of El and a ball with radius 1/2 at which D ] ( x ) ( u l ) is large and the increments in the direction u l are uniformly controlled (the real difficulty comes at this point; keeping this control is enabled by an involved estimate of behaviour of derivatives in the plane), and we continue in a similar way requiring now that u2 is close to u t, etc. The limit of the sequence x,, ~ E,, is the required point. Even from this rough description it should be clear that this approach shows that every slice S ( D f ( X ) , u, 6) (u 6 X) of the set of G~teaux derivatives of f contains a Fr6chet derivative. One of the main difficulties in proving Fr6chet differentiability results, say, for mappings of e2 to finite-dimensional spaces is that the analogous slicing statement is false: by a (complicated) example of [46] there is a Lipschitz mapping f = ( f l , f2, f3):e2 ~ / K 3 such that Df~ (el) + D I~(e2) + D II~(e3) = 0 at every point of Fr6chet differentiability of f , but not at every point of G~teaux differentiability. (Except for understandable misprints, this example, by a computer quirk, uses the meaningless symbol "_~" for (llzrozll/r,,,).) Any attempt to prove Fr6chet differentiability almost everywhere (or even existence of a common point of differentiability of finitely many real-valued functions) is greatly hampered by the fact that there may exist slices of the set of G~teaux derivatives of f containing no Fr6chet derivative. This, however, cannot happen for convex functions. The reason behind this is that they are regular in the following sense: a mapping f : X --+ Y is called regular at a point x if for every v 6 X for which the directional derivative f ' ( x , v) exists, limt~0 ,f(.~+1(,+v))-f(:~+1,) = f ' ( x v) uniformly in u with Ilull ~< 1 The key statement on Fr6chet differentiability of Lipschitz mappings with respect to F-null sets says (see [31]): if L is a norm separable subspace of the space of linear operators between separable Banach spaces X and Y, then every Lipschitz mapping f : X ~ Y is Fr~chet differentiable at F-almost every point of the set at which it is regular, G6teaux differentiable and its G6teaux derivative belongs to L. The proof is hard and draws on much of what has been done before. The basic new ingredient comes from the classical descriptive set theory: assuming, for simplicity, that f is G~teaux differentiable/-' almost everywhere, we observe

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that the mapping y ---> D f o y, being a Borel measurable mapping between complete separable metric spaces F (X) and L l(~', L), becomes continuous when restricted to a suitable residual set. To return to general Lipschitz mappings, we observe that the sets of points of their irregularity are o--porous. Hence every Lipschitz mapping o f f : X --+ R is Fr~chet differentiable F-almost everywhere if and only if every or-porous set is F-null. The condition of a - p o r o u s sets being null does not hold in s (as we will see in 4.2.3, not all Lipschitz f :s --+ ]~ are Fr6chet differentiable F - a l m o s t everywhere) but it can be proved in spaces whose structure is similar to co (for example, for spaces containing an asymptotically co sequence of finite co-dimensional subspaces). The basic method of avoiding porous sets is to modify a given y ~ F (X) close to a point at which it belongs to a given porous set so that it passes through a hole. Unfortunately, the resulting sequence of so modified ?,,, 6 F (X) may not converge (in the space F ( X ) ) . However, in the presence of a co structure we can make the modification on disjoint sets of coordinates and so achieve the convergence. These argument then give that if X is a subspace o f co, or a space C ( K ) with K countable compact, or the Tsirelson space, then all the tr-porous subsets o f X are F-null; hence all real-valued Lipschitz functions on these spaces are Frdchet differentiable F-almost everywhere. In fact, if X is a subspace of co, or C ( K ) with K countable compact, then the space of bounded linear operators from X to any RNP space Y is separable, and so every Lipschitz mapping between such spaces is Fr6chet differentiable F - a l m o s t everywhere. 4.2.3. Mean value estimates One of the important applications of derivatives or their generalizations is their use to estimate the increment of a function. The model statement is Lebesgue's variant of the fundamental theorem of calculus saying that for a real-valued Lipschitz function f of one real variable f ( b ) - f ( a ) -- f l ' f ' ( t ) d t and its corollary, the mean value estimate, that for every e > 0 there is t 6 [a,b] such that f ' ( t ) ( b - a ) > f ( b ) f (a) - e. For real-valued Lipschitz functions on a Banach space X one cannot expect that a point of differentiability can be found on the segment [a, b], and so the mean value estimate either uses a point of differentiability close to [a, b] or replaces the derivative by its generalization (this approach will not be used here). The mean value estimate for G~teaux derivatives follows immediately from the fact that every Haar null set is null with respect to linear measure on a dense set of lines. In fact this gives a stronger statement: if X is separable, G C X is open, N C X is Haar null, and f : G --+ It~ is Lipschitz, then f o r every segment [a, b] C G and every e > 0 there is x ~ G \ N such that D f ( x ) ( b - a) > f (b) - f (a) - e. Since no almost everywhere result is known for Fr6chet derivatives, the mean value estimate for them is proved by following more carefully the construction of points of differentiability: if X is an Asplund space, G C X is open, and f : G ~ R is Lipschitz, then f o r every segment [a, b] C G and every e > 0 there is x ~ G at which f is Fr~chet differentiable such that D f (x)(b - a) > f ( b ) - f ( a ) - e. (As in the existence result, it suffices to assume that the weak* closure of the set of G~teaux derivatives of f is norm separable.) The mean value estimate may be used to show that the set of points of Fr6chet differentiability of these mappings cannot be too small: if any of its projections on I~ were not of full outer Lebesgue measure, we would find a non-constant everywhere differentiable Lipschitz function q9 on R having derivative zero at the projection of every point of Fr6chet differentiability of f ; adding to

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f a large multiple of the composition of 99 and the projection would produce a function violating the mean value estimate. If X is separable, this shows that any one-dimensional projection of the set of points of Fr6chet differentiability is of full Lebesgue measure (since it is measurable). It is not known if an analogous statement holds also for two-dimensional projections. A higher-dimensional version of the mean value statement for Fr6chet derivative of a mapping f of X to a finite-dimensional space may be understood as the statement that every slice of the set of G~.teaux derivatives of f contains a Fr6chet derivative. This holds for mappings which are Fr6chet differentiable F-almost everywhere. (Basically, one considers a y representing a small finite-dimensional parallelepiped on which the mapping is well approximated by a linear mapping belonging to the slice; for a slight modification of y one gets Fr6chet differentiability/z-almost everywhere and, by the divergence theorem, the mean of the derivative changes only as little as we wish.) Because of this and of the example of [46] (which was already mentioned above), we see that Fr6chet differentiability F-almost everywhere is false for real-valued Lipschitz functions on ~2. 4.2.4. A l m o s t Frdchet derivative It has been already mentioned that for the Lipschitz isomorphism problem notions of derivatives weaker than Fr6chet derivative may be pertinent. A function f : X ~ Y is called a l m o s t Frdchet differentiable is for every e > 0 there are x 6 X and a bounded linear operator T (both x and T may depend on e) such that lim suPll,ll__,0 IIf(x + u) - f (x) - T ( u ) l l / l l u l l < ~. In [29] these derivatives were shown to exist for mappings of supper-reflexive spaces to finite-dimensional spaces (by a rather involved proof). This result was extended in [22], with a more transparent proof, to the case of asymptotically uniformly smooth spaces; this paper should be consulted for details and applications. 4.2.5. Weak* derivative For a Lipschitz mapping f of a separable Banach space X to the dual of a separable space Y one defines the weak* directional derivative of f at x in direction u as the weak* limit, as t ~ 0, of f~x+1u)-,f~) The weak* G~teaux different tiability of f at x is defined by requiring that this mapping be linear in u. The existence results for G~teaux derivatives of Lipschitz functions hold also in this setting (and do not need any RNP requirement). Of course, some of the properties of G~teaux derivatives are lost; in particular, the weak* Gfiteaux derivative of a Lipschitz isomorphism may well be zero at some points. However, mean value estimates still hold, so these derivatives are not trivial; and, starting from [32] and [ 19] have been successfully used to study the Lipschitz isomorphism problems for spaces without Radon-Nikod3~m property. 4.2.6. Metric derivative The standard example of a nowhere differentiable Lipschitz mapping of (0, 1) to Ll(0, 1), given by f :x ~ XCo,x) where XE denotes the indicator function of the set E, is an isometry. This is not just a chance, since the one-dimensional case of the following result due to Kirchheim [27] says that every Lipschitz mapping of (0, 1) to a metric space locally (near to a.e. point) multiplies the distance by a constant as if it were differentiable (no RNP type condition on of the range is needed). I f f is a Lipschitz m a p p i n g o f an open subset o f R" to a metric space, then f o r a.e. x ~ I~'~ there is a s e m i n o r m I1" IIx on ~ " such that limto0 Q(x + tu, x + t v ) / t = Ilu - vllx f o r all u, v ~ R n. (For an

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application see 1.1 .) A new proof of this statement, which relates it to differentiability of Banach space valued mappings has been found in [ 1]. We may assume that the target is the dual of a separable Banach space. Then f is weak* differentiable almost everywhere, and it is natural to assume that Ilullx -- IIO;(x)(u)ll is the required seminorm; this can in fact be shown by decomposing R/' into countably many sets in which the weak* derivative does not oscillate much (the oscillation is measured in a metric metrizing the weak* topology of a ball in E) and using the density theorem together with mean value estimates.

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[22] W.B. Johnson, J. Lindenstrauss, D. Preiss and G. Schechtman, Almost Frdchet differentiability of Lipschitz mappings between infinite-dimensional Banach spaces, Proc. London Math. Soc. (3) 84 (2002), 711-746. [231 O. Kalenda, Stegall compact spaces which are notfragmentable, Topology Appl. 96 (2) (1999), 121-132.

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