DIFFERENTIABILITY OF CONVEX FUNCTIONS AND ASPLUND SPACES

1995.15(2):171-179

DIFFERENTIABILITY OF CONVEX FUNCTIONS AND ASPLUND SPACES· Cheng Lixin Math. Section, Dept. Abstract

<<<.i..fIr)

0/ Public Courses,

Zhang Feng <~Jtt.)

Jianghan Petro. Inst;

t

Shoshi 434102 ,China.

Characterizations of differentiability are obtained for continuous convex functions de-

fined on nonempty open convex sets of Banach spaces as a generalization and application of a mumber of mathematicians several years effort, and a characteristic theorem is given for Banach spaces which are (weak) Asplund spaces. Key words

1

convex function, differentiability, Banach space t Asplund space.

Introduction The following theorem, which was established by a number of mathematicians [see,

for instance, [3JJ is now well known, and was the motivation for the results presented in this note.

Theorem

f

is a con-

Xo

E E with

Suppose that E is a Banach space with dim E~2. Suppose that

tinuous, positive homogeneous and sublinear functional defined on E and that I(xo) >0; then i)

f

is Gateaux differentiable at

%0

if and only if, for each yE E 1(%0) -

lim

,....0+ ••

sup .:jI!:"

-esnv.,(zo·")

1("

t

v)

--=-II-u---v""":":"'II--

=

0

where Uy(xo,r)=span{xo,y} nU(xo,r) and U(xo,r) is for the ball in E centered at

%0

with radius r sS denotes the level set {xEE,!(x)=!(xo)}. ii)

f

is Frechet differentiable at

lim

,--0+ ..

A real-valued function

f

sup

..."

.%0

if and only if

-e snv (%0.")

-I(~) lIu-vll =0

1(%0)

defined on a convex subset D of a Banach space E is said to be

• Received Mar. 15,1993. revised Sep. 6,1993. Supported by • foundation of COOC

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ACT A MATHEMATICA SCIENTIA

convex provided f
(l-A)y)~Af(x)+(l-A)f(Y)

Vol. 15

for all x,yE D and

O~A~l;

the

subdifferential map af of f on D is defined by af(x) = {z '

E E· ;

< a:' ,y -

x >~ fey) -

f(x) for all y

E D}

The letter E will always be a Banach space and E· its dual; f denotes a real valued convex continuous function defined on a nonempty open convex subset D of the space E and

S (xo ,f) denotes the level set {x ED; f (x) = f (xo) }; and it is also denoted simply by S whenever no confusion results. U (xo , r) is for the open ball in E centered at a:« and with radius r, The notation below follows from [7]and [8J: We say that

E E· such that

I

is Gateaux differentiable at xED provided that there exists a unique a:'

< a:' ,y -

for all y E D

x >~ fey) - fCx')

or equivalently, that af(x) is a singleton;

f is called Frechet differentiable at xE D provided that there exists a unique

a:'

E E·

such that for any £>0 there is 0>0 satisfying

o ~ fey) whenever yE D with II y

r-

- fCx)

x II

-< x" ,y -

x >~

£

~I y - x

II

<0.

The following properties can be found in [7,8,5, 11 J Proposition

f is continuous and convex on the nonempty open convex

Suppose that

set D, then a)

I

is locally Lipschitzian on D [7 .p, 4J;

b) a/(x) is a nonempty weak

*

compact convex subset of E* for each xE D [7 -p. 7J;

c) af is locally bounded on D [7, p.4J; d) For any xE D and yE E [11, P. 397J

d" f(x)(y)

= lim f(x + ty)t -

f(x)

=

max{< a:'

=

f(x)

=

min{<

~y >;x' E

af(x)}

E

af(x)}

1-0+

d- f(x)(y)

lim f(x 1-0-

e)

+ ty) t

-

a:' ,y>;x'

f is Frechet [Gateaux] differentiable at xE D if and only if there exists a selection ¢

for the subdi£ferential map

at,

which is norm-to-norm [rnorm-to-weak ' ] continuous at x

[[7, p. 19J, or [8JJ; f)

f

is Frechet [Gateaux] differentiable at

Xo

if and only if every selection for

at on D

is norm-to-norm [norm-to-weak ' ] continuous at xo[5, p. 134 &. p. 148J. This note presents an extension of above Theorem to general continuous convex functions defined on nonempty open convex sets of Banach spaces, and a characteristic theorem of Banach spaces which are [weak] Asplund spaces.

2

Differentiability of Convex Functions We start with the following theorem:

173

Cheng &. Zhang: DIFFERENTIABILITY OF CONVEX FUNCTIONS

No.2

Theorem 1

f

Suppose that

is a continuous convex function defined on a nonempty

open convex set DCE with dim E~2, and suppose xoED with f(xo»inf{f(x);xED}; then

f

is Gateaux differentiable at Xo if and only if both the following conditions hold.

i) The xo-direction direvative of f exists . that is, d" f(xo)(xo)=d- f(xo)(xo);

f(~) lIu-vll

f(xo) -

ii) lim

sup

r-o+ •. .,esnu.,,(,zo·r)

(1)

=0

.,:;a!:t'

for any y E E, where S denotes the level set S (xo, f) and U ~ (xo, r) = span {xo, y} U(xo,r).

Proof

Sufficiency. Without loss of generality we can assume that 0 ED, f (0) =

f(xo) = 1 [otherwise, we may replace the function

where

fl(X)

=

k[f(x

+ Xl) -

f

/1 and Ds ,

and the set D by and D 1

!(x1 ) ]

=

D -

n

°and

resp. ,

Xl

the point Xl being chosen so that f(Xl)OJ Suppose, that

f

is not Gateaux differentiable at Xo; then af(xo) is not a singleton.

First, we claim that af(xo) can not be represented by a segment lying in a ray of the

x; E E and [a,b]C

dual E· , or equivalently, af(xo)#[ax; , hx; JCE· for any non-zero

[0,(0). Suppose af(xo)=[a.x; , hx; J==1 for some non-zero x; EE· and [a, b]C[O,oo) with a = c>

= {xEE;
°and let H

,x>=O}, then for each yEE there exist a unique aER and a unique hER

such that y=ayc +h. Noting Proposition d , we observe

=

bCa , if a ~ 0 { aca , if a < 0

d" f(xo)(Y)

=

max{< x:' ,y >;x· E I}

d- f(xo)(Y)

=

min{< a:' ,y >;x· E I} = ~. lbca, If a

(2)

and

since f(O)=O, f(xo)=l, for any f(Ax o )

and

=

f(Ax o

+ (1

1 = f(xo) = !().().,-lxo ) )

(aca,

O<,l~l,

we have

-

il) • 0) ~ Af(xo)

=

f(,l(A-1xo)

f(p.xo) ~ pf(xo)

=

+

+ (1 -

if

a ~

0

<

0

il)f(O)

=

(3)

A

(4)

(1-).,)0) ~ ).,f().-l x o) , that is

p.

for p. ~ 1

(5)

Hypothesis i) of sufficiency and both the inequalities (4), (5) imply

1 ~ lim f(x o

+ txo) -

~o-

= d" f(xo)(xo)

t

f(x o) =

s: f(xo)(x o)

= lim f(x o + txo) 1-0+

t

-

f(xo)

~1

From above inequalities and (2) or (3), we also observe that XoE H, and that there exist Q o

+ho and inequalities (2) and (3) in turn imply f(x o ) (x o) = bca; > aca o = d- f(xo) (xo)

E R+ and ho E.t H with Xo = aoyo d"

and this is a contradiction.

(6)

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ACTA MATHEMATICA SCIENTIA

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By the fact we have just proved, we can choose x; . x; E af (xo) such that

{xEE; =O}7-{xEE;=O} and we can choose again zE {xEE;
II z II

+x; ,x>=O} with

=1 such that

==d>O

(7)

Since D is open and Xo ED, for such z and for all sufficiently small t>O, say,

t~ro for

some

ro>O, we have xo±tzED. Therefore, by definition of a/(xo) for all t with O~t~ro f(xo

and

+ tz) -

1 = f(x c

= ft x,

1

f(xo - tz) -

+

tz) -

ts:) -

-

>= td t.z >= td

f(x o ) ~< x; ,tz [(xo) ~< x:* , -

(8)

(9)

For each such t and for sE[O,t], define

=

u(s)

+ tx

+ sz) = ,i(s)

Xo

f(xo

= p(s)(x

x - tz 0 ) Xc sz

v(s) = f(

where ,i(s) =

• (xo

+ tz) , tz),

o-

f(xo+sz)-l~(l +sd)-l~l ,pes) = f(xo -SZ)-l~(1 +Sd)-l~1,

!(u(o) !(u(t)

The continuity of

+ tz) ~ 1 + td and (by(4») =!(A(t)(x o + tz') ~ A(t) • j(xo + tz)

and note

=j(xc,

f says that for each

there exists O
O
= ,i(a,)(x

u(a,)

= 1

o

+ tz)

E S

(10)

tz) E S

(11)

and similarly, there exists O<{jt
= p(!3t)(x

o-

Next, we show thatthere exists 0>0 such that 1~

alt ~ ~ and 1 ~ Pit ~ 0 for all sufficiently small

t

>0

Suppose that there exists some sequence {t.} in R+ with t, -0 such that at. t,

=

(12) a" -0. t rt

Since

f

is locally Lipschitzian on D [Proposition d)], by noting u(art ) =,i(a,.) (xo+t"z) .....j(xo.)-lxo

= xo,

there exists M>O (a local Lipschitz constant of

ciently large

n~l

f
1

f(xo

a,.

~ M II (x o

+ t"z)

X

o

= M II Xo

xo) such that for all suffi-

- !(u(a.))

art

+ t.z) -

u(a.) II = M II (l -

,l(a.))(xo

~

= M II

f around

+ t.z)

II

~

+ t"z II ( 1 -

,l<~)

aft

)=

+ t.z II (f(x o + a"z) a..

M

II

X

o

+ t"z II [ A-I (art) - 1] • ,i(a.) ~

- !(Xo)) • ,l(a.) -

M II Xo II d+ !(xo)(z)

<

00

On the other hand f(xo

+ t.z) a"

-

1 = ~ , f(xo aft

+ t.z) t;

-

1~ ~ •d _

00

art

this contradiction explains that the first inequality of (12) is true; and in a like manner, we

175

Cheng s, Zhang: DIFFERENTIABILITY OF CONVEX FUNCTIONS

No.2

may imply that the second inequality of (12) also holds for some

~>O.

Finally, letting ut=u(a,) [ES] and v,=v(,8,)[ES]) and letting M>O be a local Lipschitz constant of f around xo, for all sufficiently small t>O, we see

o<

u, -

~

1 with limA(a,) limp(p,) = 1 '-0 '-0

p([3,) o,

(14)

IA(at ) - p(P,) I II Xo II + (A(a,) + p(P,)t I % 1\ 2Mt II X o II + 2t = 2(M II Xo II + i»

~

~

f around

and [by the local Lipschitz condition of feu,

(13)

I = A(a,)p({3t) If(x o - (3,z) - fCx; + atz) I ~ 2Mt II = II A(a,)(xo + tz) - p(P)(xo - t z') II

IA(a,) 1\

p(,8,)

A(a,) ,

(15)

Xo again]

~ v,) = f().(a,) ~ p(P,) Xo + ~ ().(a,) - pCP,»tz)

~

f().Ca,)

~

).Ca,)

~ 2[1

Note u, v, E span {xo ,z}

o = lim

1

pCP,) )tz

(),Ca,) -

II

t

2 2

1

1

(2)

• lit

+M

2 2

t

nS, thus by the hypothesis and above inequalities we obtain ~ v)

!(x o) - I(u

sup

~ ,~~

+M

~

+ at • d + 1 + p,d] + Mtr ~ 1 + d

II U

r-oo •. t'EsnU!'(.ro.r)

.

+ M II

pCP,) Xo)

~ pCP,) ICxo)

1

(9). (10)

~

1 - [1 + 2(M II

-

~ . lit + M X o II + 1)t

v 2

f(xo) _ feU,

~ ,~~

II

t=]

d • = 2(M

II

II

in!

Xo

~

u, - v,

+ 1) >

II

(6)

~ v,)

I

0

and this is a contradiction. Necessity. Suppose that fis Gateaux differentiable at xoED, then a!(xo)(=x·) is a singleton. By Proposition d),d+ f(xo)(xo)==d- !exo) ( x o) , that is, i) being true. We may assume [Proposition e) ] that rp is a selection for the subdifferential map af on

D, which is norm-weak * continuous at Xo. For any yEE, and for all u,vESnUy(xo,r) with u¢v, let

O -/'

v-u

1

w= II v-u II ,t=2 II v-u II

!(xo) -

II

~

feu

u - v

+ v) II

2

then

_ l/(u) - !(u 2 t

-

1

1

+ tw) -/' _ 1- < ~

=-2- 2 <9'

since = uous at

Xo

it

2

,>

r1"1" ",U),W

(17)

U(U)- f(v)- J~O and cp is norm-to-weak' contin-

and since w is in span {xo' y} with

II

w

II

= 1, we see that the right hand side of

the last inequality of (17) must tend to zero whenever u,vE S and u,~x, and which com-

176

ACTA MATHEMATICA SCIENTIA

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pletes our proof. Remark

From the proof of the sufficiency part in Theorerm 1 after inequality (6),

we observe that we can substitute the condition that the function g: defined by g (x· )

II

II

a:'

EE*, is a constant on

«a:'

a/(xo) for hypothesis i) of Theorem

=

1.

The following example shows that in the converse part of Theorem 1, hypothesis i) is also essential. Let E=R" and define the continuous convex function! on the right open

Example

half plane D= {(x,y)ER 2 ;x>O} by

E D and

!(x,Y) = x , (x,)')

x ~

1; = x 2 , otherwise

let (xo, Yo) = (1 , 1) then the level set S = { (1 -s ); y E R}. Clearly, for all (Xl' Yl), (x:, Y:) E S, f(x!

~x: ,Y:~Y:) = f(xo ,Yo) = 1,

and f is not differentiable at (1,1) since {(r,

0) ;rE [1, 2J }Ca!(l, 1). Theorem 2 differentiable at

With the same Xo

f, E , D , Sand

=

d- !(xo) (xo),

feu ~ v) -~Il-u---v-II--

!(xo) 1-0+ •• toE

E D as in Theorem 1; then f is F rechet

if and only if

i) d" I(x o) (x o ) ii) lim

Xo

sup

snu(.rc · ,.)

= o.

(18)

W::;Cto

Proof

Necessity. Apply Frechet differentiability of

I at

Xo

in place of Gateaux differ-

f is

entiability in the proof of necessity in Theorem 1 and recall [Proposition e) ] that Frechet differentiable at

Xo

if and only if there is a selection (ffor

alon D which

is norm-to-

norm continuous at xo, so this direction is immediately proved. Sufficiency. By the hypothesis and Theorem 1,

!

is Gateaux differentiable at

Zo.

As

before, we may assume that OED,!(O)=O and !(xo)=l. Suppose, to the contrary, that £>0, {xlI}CD with

f

II XO-X II <1;. II

n~

is not F rechet differentiable at Zo. Then there exist

x: Ea!(x.)

and

£>0 and for n = 1, 2, ••. , where x; E


,xo>~,

a! (xo ).

such that

II x; -x: II ~£ for

some

By Proposition f), one must have

and inequalities (4), (5) and Proposition d) further imply

d" I(xo) (xo) =d- !(zo) (zo) = 1 = , which in turn imply

z:

= (x;

+ x; ) II x; + x,;

II -} · II x;

1\

E {x * E E· ; II

II x;

-

z,.*

a:"

II

II

x;

II >

O}, (19)

and for all integers

n~l

II

~c

(20)

and for some c>O.. Combining with the Bishop-Phelps Parallel Hyper-

plane Lemma [see, for instance 4,p.

2JJ

EHJI==.{xEE,

II

=O} with

together, we observe that there exist 0>0 and y.

yll

[=
II

=1 such that

-y,,>J~20' forn=1,2,···

(21)

Cheng s, Zhang: DIFFERENTIABILITY OF CONVEX FUNCTIONS

No.2

177

Thus, for any l~a>O, for all t~.!!... and sufficiently large n with x ..±ty..,xo+ty" ED f(x"

n

+ ty,,)

=f(xo

+ ty.) + [f(x. + ty.) -

~f(xo) +< x; ,ty" >- M II Xo It», - ty..)

~f(x,.) ~1

+< x ..- ,

+ 20t

- tv,

>=

+ ty,,)]

f(xo

II ~ 1 +

x;

-

f(xo)

2C1t -

~ n M

+< x; ,ty. >+ [f(x..)

-

f(x o ) ]

M

~ n'"

-

where M denotes a local Lipschitz constant of f around xo, that is, for all sufficiently large n and

l~t~.!!.. n n

with x.±ty.ED, f(x.. ± ty,,) ~ 1

+ at

(22)

By an arguement which is much like the one in Proof of sufficiency in Theorem 1, for

o
each such n, there exist

u..

and

v ..

x .. f(x"

=

f(

such that

1 + -;;y" 1 + A..y,,) = p,.(x.. + -;y.) E 5,

x; -

=

o<,u,.~l n

and

1 -;;y,.

x .. - J.t,.y.

)

= q,.(x.. -

1 -Y.)

n

E5

and there exists 0>0 such that 1 ~ A.-n ~ 8

and

1 ~ /-4.n ~ 8 for all n ~ 1.

and by noting (22) we obtain [for all sufficiently large n~l] (23)

and q;l = f(x.. -

So that (by local Lipschitz condition of If(xo) -

Ip. -

[(x.)

~ M I p.,.

II u;

- v .. 11=

II

J.t,.y,.) -

+ A,. I II y.. II

(P.. -

q,.)x,.

+

~2M~+l.. n n

(1

f around

I~ M 2" n

q,.1 = P. • q.lf(x.. -

+ au; ~ 1 + -n · 8

p.,.y.) ~ 1

f(x.

xo)

+

,l,.y.) I

1

~ 2M • n

1

-(P.. n

+ q,.)y.1I

~

Ip.. - q.11I x.1I +l.. n

(24)

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Vol. 15

ACTA MA THEMATICA SCIENTIA

t

+ ~z (by ~ p" + q" + M + M% ~ ~ 2 n n%~

~ f(P.

q·x.)

2

4-. t q.f(x.) + ~2

(4))

1

+ M(M + 1)

a

nZ

1 +-8 n

It follows from the hypothesis and the inequalities above

o=

lim

sup

f(x o) - f(u

II U

,...0+ •• "ESnU(zo·I')

v

-

t

v)

f(xo) _ f(u.

~ li~

II

.~"

=

Ii

1 - «1

m

+ .!a)-l + M(M + l)n-%) n 2(M II x; II + 1)n1

~ v.)

II Un - v. II 1

=

l'

va

Z M II Xo II + 1

>

0

and this is a contradiction which completes the proof. A Banach space E is called an Asplund [a weak Asplund] space provided every contiuous convex function defined on a nonernpty convex subset D of E is Frechet [Gateaux] differentiable on a dense G, subset of D. For this topics, we can refer to [7J, [5J and recently, a survey paper [9J. Let D be a nonernpty open convex set of the Banach space E, and CD the cone of all continuous convex functions defined on D, and let V D denote the cone of all real valued nonnegative functions defined on D and vanishing on some dense G,-subsets of D. We define the transforms F and G on CD as follows. !(x) F(f)(x)=

lim

sup

II

,...0+ ..."ESnU(z.r)

-:¢"

!(u

u - v

!(x) _ G(f)(x)= sup lim :fEE

II U

sup

r-o+ ... vE s n u.1( z o . '

)

If?:"

where S denotes the level set {z ED;

f

+2 v)

II

,x E D

!(" + v) -

(z)

v

II

2

=!

(~F(f)(x)), xED (x )} and

U y (x, r) is for span

{x,y} nU(x,r).

The following characterization of (weak) Asplund spaces can be deduced from Theorems 1 and 2.

Theorem 3

A Banach space E is an Asplund (weak Asplund) space if and only if for

each nonempty convex open subset D of E and for each fEC D , F(f) [G(f), resp. ]belongs to V D.

Proof

By Theorem 2 (Theorem 1), it suffices to show sufficiency part.

Let usrecall Kenderov's Theorem [see, [6]or [5, p. 135JJ; For any continuous convex function

f defined on a nonempty open convex set D of a

Banach ~P~H"~ F;. th~r~ ~xi~t~ dpn~ G,. subset A of

n such that at each

point .r of whirh thp

function g defined by g(x·)= II x" II for x" EE· , is a constant on aj(x). Suppose that D is a nonernpty convex open set in the space E, and suppose F (f)

Cheng &. Zhang: DIFFERENTIABILITY OF CONVEX FUNCTIONS

No.2

179

[G(f)]EV Dfor each fECD, or equivalently. F(f) (z)

=

lim

...-0+ -.-oESnU(z.r) q-o

(G(f) (z') = sup lim ~E E

-f(~) II u-vll

f(x o)

sup

f(xo) -

sup

,....0+ -.-oEsnu,,(z.r)

f("

II " -

v

t

II

v)

,resp).

~-o

vanishes in a dense C, subset B of D. We need only show that f is Freehet [Gateaux, resp. ] differentiable on some dense G, subset of D. Let A denote a dense G1 subset D, at each point x of which (by Kenderov's Theorem) the function g[g(x· ) =

II a:" II ]

is a constant on a!(x) , and let C be the dense open set

{xE D; f Cx) >inff} of D. Then by combining with the assumptions, with Theorem 2

[Thereom 1, resp. ] and with Remark deduced from Theorem 1, we obtain that

! is

Frechet [Gateaux] differentiable on the dense G, subset AnBnC of D. Acknowlegdement

The author wants to express his special thanks to professors R. R.

Phelps and I. Namioka for their helpful conversations on this note, and to Professor E. L. Stout, Chairman of Dept. of Math. , Univ. of Washington, for his hospitality when the author was studying there. References 1

Chen Daoqi. A sufficiency condition for the support functional being a unique. Acta. Math. Sinica. 1982,25(3):302--305.

2

Cheng Loon. Two notes on smoothness of Banach spaces. J. Math. Res. Exp. , 1989,9(2): 175-176.

3

Cheng Lixin. Li Janhua , Nan Chaoxun, Gateanx and Frechet differentiability of .continuous gauge

4

Diestel J. Geometry of Banach spaces. Lect, Notes in Math. , Nr, 485, Springer-Verlag. 1975.

functions on Banach spaces. China: Adv. in Math. , 1991,20(3): 326-334.

J R. Convex analysis with application in differentiation of convex functions. Res. Notes in

5

Giles

6

Kenderov S P. The set-valued monotone mappings are almost everywhere single-valued. C. R. Acad.

Math. , Nr, 58, Boston-London-Melboume , 1982. Bulgare Sci. , 1974,27: 1173-1175. 7

Phelps R R. Convex functions, monotone operators and differentiability. Lect, Notes in Math. , Nr,

8

Rainwater I.

1364, Springer-Verlag, 1989. Yet more on the differentiability of convex functions. Proc. Amer. Math. Soc. , 1988.

103: 773-778. 9

Wu Congxin , Cheng Loon. A survey of the differentiability of convex functions and Asplund spaces. ICBST'94, Wuhan Univ. ,1994.

10 Yu Xintai, A sufficiency condition for Frechet differentiability points of norms of Banach spaces. China: Adv. in Math. , 1986.15(3). 11

Yu Xintai. Geometric theory of Banach spaces. Shanghai: East China Teacher's Univ. Press, 1986.