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Generic Fréchet Differentiability of Convex Functions on Non-Asplund Spaces
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
214, 367]377 Ž1997.
AY975570
Generic Frechet Differentiability of Convex Functions ´ on Non-Asplund Spaces Cheng Lixin and Shi Shuzhong Nankai Institute of Mathematics, Nankai Uni¨ ersity, Tianjin, 300071, People’s Republic of China
and E. S. Lee Department of Industrial Engineering, Kansas State Uni¨ ersity, Manhattan, Kansas 66506-5101 Submitted by William F. Ames Received April 4, 1997
Let f be a continuous convex function on a Banach space E. This paper shows that every proper convex function g on E with g F f is generically Frechet ´ differentiable if and only if the image of the subdifferential map f of f has the Radon]Nikodym ´ property, and in this case it is equivalent to showing that the image of f is separable on each separable subspace of E. Q 1997 Academic Press
1. INTRODUCTION Starting with Asplund’s work w1x, the notion of an Asplund space first appeared in Namioka and Phelps w5x: A Banach space is said to be an Asplund space if every Žreal-valued. continuous convex function defined on a non-empty convex open subset of the space is Frechet differentiable on a ´ dense Gd subset of its domain. Now, it is well known that a Banach space E is an Asplund space if and only if its dual EU has the Radon]Nikodym ´ property Žsee, for instance, w6x.. However, on a non-Asplund space, one can still find many non-trivial continuous convex functions, which are Frechet ´ differentiable on a dense Gd subset of its domain. 367 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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LIXIN, SHUZHONG, AND LEE
Recently, Tang w7x obtained the following theorem: THEOREM T. Suppose that f is a con¨ ex Lipschitz function on a separable Banach space E. Then the following statements are equi¨ alent: Ži. e¨ ery continuous con¨ ex function g on E with g F f is Frechet ´ differentiable on a dense Gd subset of E; Žii. the image of the subdifferential map f of f, f Ž E . s xU g EU : U x g f Ž x ., x g E4 is separable. Giles and Sciffer w3x showed that THEOREM GS. Let f be a continuous con¨ ex function on a Banach space E. If the image of the subdifferential map f of f is separable on each separable subspace of E, then f is Frechet differentiable on a dense Gd subset ´ of E. Both theorems show that on a separable Banach space E, which is not necessarily an Asplund space, and even on any Banach space E, there exist many generically Frechet differentiable continuous convex functions. ´ In this paper we will discuss similar problems and generalize and unify both theorems on any Banach space. The final result is as follows. MAIN THEOREM. Let f be a proper lower semi-continuous Ž l.s.c.. con¨ ex function on a Banach space E and its effecti¨ e domain dom f be open. Then the following statements are equi¨ alent: Ži. e¨ ery proper l.s.c. con¨ ex function g on E with g F f is Frechet ´ differentiable on a dense Gd subset of its effecti¨ e domain dom g; Žii. the image of the subdifferential map f of f, f Ž E . s xU g EU : U x g f Ž x ., x g E4 , is separable on each separable subspace of E; Žiii. the image of the subdifferential map f of f, f Ž E ., has the Radon]Nikodym ´ property; Živ. the wU -closed con¨ ex hull of the image of the subdifferential map f of f, wU -cl cow f Ž E .x, has the Radon]Nikodym ´ property. Taking f to be the indicator function of 04 , this theorem also generalizes the classic result: A Banach space E is an Asplund space if and only if its dual EU has the Radon]Nikodym ´ property or EU is separable on each separable subspace of E. The paper is organized as follows. In the next section, we introduce the notation and recall some preliminaries. Sections 3 and 4 discuss the generic Frechet differentiability of convex functions dominated by a lower ´ semicontinuous sublinear function or a continuous convex function. Section 5 completes the proof of the Main Theorem. In the last section, some concluding remarks are made.
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2. NOTATION AND PRELIMINARIES We will always denote a Banach space by E and its dual by EU . ² ? , ? : is the dual product on EU = E. Let f : E ª R j q`4 be a proper extended U real-valued function on E. Then the subdifferential map f : E ª 2 E of f is defined by
f Ž x . s xU g EU : ; y g E, f Ž y . y f Ž x . G ² xU , y y x : 4 . The conjugate function f U : EU ª R j q`4 of f is defined by f U Ž xU . s sup ² xU , x : y f Ž x . 4 . xgE
It is easy to check that
f Ž x . s xU g EU : f Ž x . q f U Ž xU . s ² xU , x : 4 . When f is convex on E and continuous at x, then f Ž x . is non-empty. If f is lower semi-continuous Žl.s.c.. and convex, i.e., its epigraph, defined by epi f [ Ž x, a . g E = R: f Ž x . F a 4 , U
is closed and convex, then f : E ª 2 E is a maximal monotone map w6, p. 59x; and ; x g E,
f Ž x . s sup
xUgE U
² xU , x : y f U Ž xU . 4 .
In addition, in this case, f is continuous on the interior of its effective domain, int dom f, where the effective domain of f is defined by dom f [ x g E: f Ž x . - q` 4 . See w6, p. 41x. In particular, if dom f is open, then f is continuous in dom f and the graph of f, graph f [ Ž x, f Ž x . . : x g dom f 4 , becomes the boundary of epi f. For a convex function f on E, f is Frechet differentiable at x g E if and ´ only if for any « ) 0 there exists d ) 0 such that 0 F f Ž x q ty . q f Ž x y ty . y 2 f Ž x . - t « whenever 5 y 5 s 1 and t g Ž0, d .. It is also equivalent to showing that f is single-valued and norm-to-norm upper semi-continuous at x w6, p. 48x.
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LIXIN, SHUZHONG, AND LEE
Every l.s.c. sublinear function p: E ª R j q`4 is the support function of a subset of EU ; i.e., there exists a subset CU ; EU such that ; x g E,
p Ž x . s sC U Ž x . [ sup ² xU , x : . xUgC U
p is continuous on E if and only if CU is bounded. It is easy to show that we can take CU s p Ž 0 . s xU g EU : ; y g E, ² xU , y : F p Ž y . 4 , and thus pU Ž xU . s dC U Ž xU ., the indicator function of CU , which is valued 0 on CU and q` out CU . Hence, when x / 0, xU g p Ž x . m xU g C U
and
² xU , x : s p Ž x . .
A non-negative continuous sublinear function on E is called a Minkowski functional. Every Minkowski functional pC : E ª Rq associates with a unique closed convex set C ; E with 0 g int C such that ; x g E,
pC Ž x . s inf l ) 0: x g lC 4 ,
and C s x g E: p Ž x . F 1 4 . We also say that pC is the Minkowski functional generated by C. In this case, CU is the polar set of C; i.e., CU s xU g EU : ; x g C, ² xU , x : F 1 4 . Let S [ C s x g E: pC Ž x . s 14 . Then pC Ž S . consists of all support functionals xU of C at x with ² xU , x : s 1. By the Bishops]Phelps theorem w6, pp. 51]52x, the support functionals are dense in the cone of all those functionals which are bounded above on C. In particular, we have that PROPOSITION 2.1. pC Ž S . is dense in C1U [ xU g EU : sup x g C ² xU , x : s 14 and thus, pC Ž0. s cl co pC Ž S . j 044 . We will use the Minkowski functional generated by the epigraph of a l.s.c. convex function with an open dom f. In this case, we have that C s epi f and S s graph f. We need the following proposition: PROPOSITION 2.2. Let f : E ª R j q`4 be a proper l.s.c. con¨ ex function on E and continuous at x s 0 with f Ž0. s y1. Let p: E = R ª R be
371
CONVEX FUNCTIONS
the Minkowski functional generated by epi f. Then xU g f Ž x . if and only if
Ž yU , r . g p Ž x, f Ž x . .
with yU s xUrf U Ž xU .
and r s y1rf U Ž xU . ; f is Frechet differentiable at x g E if and only if p is Frechet differentiable at ´ ´ Ž x, f Ž x .. g E = R. Proof. In fact, xU g f Ž x . is equivalent to f Ž x . q f U Ž xU . s ² xU , x :; i.e., 1 U
f Ž xU .
² xU , x : y f Ž x . s p Ž x, f Ž x . . s 1,
and noting that ; Ž y, r . g epi f ,
² xU , x : y f Ž x . G ² xU , y : y f Ž y .
s² Ž xU , y1 . , Ž y, f Ž y .: G² Ž xU , y1 . , Ž y, r .: , we have that Ž xU , y1.rf U Ž xU . g pŽ x, f Ž x ... Here, we recall that f Ž0. s y1 implies that f U Ž xU . s sup ² xU , x : y f Ž x . 4 G yf Ž 0 . s 1. xgE
From this equivalent relation, it is easy to check that f is single-valued and norm-to-norm upper semi-continuous at x if and only if p with
p Ž x, f Ž x . . s Ž yU , r . g EU = R: yU s
½
xU U
² x , x: y f Ž x .
,rs
y1 U
² x , x: y f Ž x .
5
is single-valued and norm-to-norm upper semi-continuous at Ž x, f Ž x ... Let AU ; EU , x g E with x / 0 in E and a ) 0. Then we say that S Ž x, AU , a . [ xU g AU : ² xU , x : ) sAU Ž x . y a 4 is a wU -slice of AU , where sAU Ž x . [ sup xU g AU Ž x . is the support function of AU . We say that a non-empty subset AU of EU is wU-dentable, if it admits wU-slices of arbitrarily small diameter, i.e., for every « ) 0, there exist x g E with x / 0 and a ) 0 such that diam S Ž x, AU , a . - « .
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LIXIN, SHUZHONG, AND LEE
A subset AU ; EU is said to have the Radon]Nikodym ´ property ŽRNP. if Ži. every non-empty bounded subset of AU is wU -dentable. If AU is wU-closed and convex, then AU having the RNP is equivalent to saying that Žii. AU is separable on each separable subspace of E; i.e., for every separable subspace E1 of E, AU < E1 is separable. ŽSee w2, p. 91x for more equivalent conditions.. In particular, EU is said to have the RNP if every separable subspace of E has a separable dual. The following theorem is due to Kenderov Žw4x; see also w6, p. 32x; the proof was only for CU s EU , but it is suitable for any CU ; EU .: THEOREM 2.1. Assume that CU ; EU is wU-closed con ¨ ex and has the U RNP. Then for e¨ ery maximal monotone map T : E ª 2 C with int DŽT . / B, where DŽT . [ x g E: T Ž x . / B4 , there is a dense Gd subset G of DŽT . such that T is single-¨ alued and norm-to-norm upper semi-continuous at each point of G. A proper l.s.c. convex function f on E with int dom f / B is said to be generically Frechet differentiable on dom f if f is Frechet differentiable on a ´ ´ dense Gd subset G of dom f. We will say that this function has the Generic Frechet Differentiability ŽGFD.. By Theorem 2.1, f has the GFD if the ´ U w -closed convex hull of f Ž E ., denoted by wU -cl cow f Ž E .x, has the RNP, because f is maximal monotone. The proof of the following proposition can be found in w6, pp. 25]26x. PROPOSITION 2.3. Suppose that AU ; EU is bounded and does not admit w -slices of arbitrarily small diameter. Then the continuous support function sAU is nowhere Frechet differentiable. ´ U
3. GFD OF CONVEX FUNCTIONS DOMINATED BY A L.S.C. SUBLINEAR FUNCTION In this section, we will seek a necessary and sufficient condition for the GFD of convex functions dominated by a l.s.c. sublinear function. THEOREM 3.1. Suppose that p: E ª R j q`4 is a l.s.c. sublinear function. Then the following statements are equi¨ alent: Ži. Žii. Žiii.
e¨ ery proper l.s.c. con¨ ex function g with g F p has the GFD; e¨ ery l.s.c. sublinear function g with g F p has the GFD; pŽ0. has the RNP.
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If p is a Minkowski functional, then they are also equi¨ alent to Živ. e¨ ery Minkowski functional g with g F p has the GFD; Žv. for S s x g E: pŽ x . s 14 , pŽ S . is separable on each separable subspace of E. Proof. Let CU [ pŽ0. s xU g X U : ; x g E, ² xU , x : F pŽ x .4 . Then Ž p x . s sC U Ž x .. Suppose that Ži. holds; i.e., every proper l.s.c. convex function g with g F p has the GFD. Then Žii. and Živ. hold. If CU has no RNP, then there exists a bounded subset AU ; CU such that AU does not admit wU-slices of arbitrarily small diameter. Let sAU be the support function of AU . Then sAU is a continuous convex function on E. By definition, we have that sAU F sC U s p, and by Proposition 2.3, sAU is nowhere Frechet differentiable, which leads to a contradiction. Hence, Žii. ´ Ž . implies iii . In the case that p is a Minkowski functional, we can take g 1 [ sup sAU , 04 . Then g 1 is also a Minkowski functional. But it has no the GDF. This contradiction shows that Živ. implies Žiii.. Conversely, if Žiii. holds, i.e., CU has the RNP, then we can show that for every l.s.c. convex function g with g F p, the image of g is in CU , i.e., U g: E ª 2 C and then, by Theorem 2.1, g has the GFD. In fact, if there exists xU g g Ž x . with xU f CU , then by the Separation Theorem, there exists z g E and d ) 0 such that ² xU , z : ) d q sC U Ž z . s d q p Ž z . , which implies z g dom p. Since xU g g Ž x ., we have that ; y g E, g Ž y . y g Ž x . G ² xU , y y x : . Hence, taking y s kz, we obtain that g Ž kz . y g Ž x . G ² xU , kz : y ² xU , x : ) k d q p Ž kz . y ² xU , x : , which implies g Ž kz . ) pŽ kz . for all sufficiently large k ) 0 and leads to a contradiction with the definition of g. Thus, Žiii. implies Ži.. Finally, if p is a Minkowski functional, then, by Proposition 2.1, it is easy to check that Žiii. is equivalent to Žv.. COROLLARY 3.1.
E is an Asplund space if and only if EU has the RNP.
In this case, we can take CU s EU . Then pC U s d04 and pŽ0. s EU . Theorem 3.1 may be modified for convex functions ‘‘locally’’ dominated by a l.s.c. sublinear function. THEOREM 3.2. Suppose that p: E ª R j q`4 is a l.s.c. sublinear function. Gp is the collection of all l.s.c. con¨ ex functions g satisfying that for any x g dom g, there exists d x ) 0 and c x ) 0 such that ; y g dom g with 5 y y x 5 - d x , g Ž y . y g Ž x . F cx pŽ y y x . . Then e¨ ery g g Gp has the GFD if and only if pŽ0. has the RNP.
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LIXIN, SHUZHONG, AND LEE
Proof. Let CU [ pŽ0.. Then pŽ x . s sC U Ž x .. Suppose that every g g Gp has the GFD. If CU has no RNP, then there exists a bounded subset AU ; CU such that AU does not admit wU-slices of arbitrarily small diameter. Let sAU be the support function of AU . Then sAU is a continuous convex function on E. By definition, we have that sAU F sC U s p, and then, by the sublinearity of sAU , ; x, y g E,
sAU Ž y . y sAU Ž x . F sAU Ž y y x . F sC U Ž y y x . .
Hence, sAU g Gp . Again by Proposition 2.2, sAU is nowhere Frechet ´ differentiable, which leads to a contradiction. Conversely, it is easy to check that for any g g Gp , g Ž E . ; CNU [ D ng N n pŽ0.. Since 0 g pŽ0., pŽ0. has the RNP if and only if CNU has the RNP. Again by Theorem 2.1, g has the GFD. COROLLARY 3.2. Assume that p is a Frechet differentiable Ž not necessarily ´ . equi¨ alent norm. Then e¨ ery p-continuous con¨ ex function has the GFD. This is because we can check that pŽ0. has the RNP and that every p-continuous convex function is locally p-Lipschitz as in the case that p is the proper norm of E.
4. GFD OF CONVEX FUNCTIONS DOMINATED BY A CONTINUOUS CONVEX FUNCTION Theorem 3.1 may hold in the case where a l.s.c. sublinear function is replaced by a proper l.s.c. convex function with an open effective domain. This convex function is continuous on its domain. Let f be a proper l.s.c. convex function on E with an open dom f, and CU s f Ž E .. Then for pŽ x . s sup xU g C U ² xU , x :, we have that pŽ0. s wU -cl cow f Ž E .x. Without loss of generality, we will always assume that 0 g int dom f and f Ž0. s y1. PROPOSITION 4.1. Under the abo¨ e assumption, the following inequality holds: ; x g E,
f Ž x . y f Ž 0. F p Ž x . .
Proof. If x s 0, the inequality is trivial. If x / 0 and x g dom f, then f is continuous at x and f Ž x . is non-empty. Thus, for xU g f Ž x ., f Ž x . y f Ž 0 . F ² xU , x : F sup ² xU , x : s p Ž x . . xUgC U
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If x f dom f, i.e., f Ž x . s q`, then we show that pŽ x . s q`. In fact, in this case, there exists l x g Ž0, 1. such that l x s sup l g Ž0, 1.: l x g dom f 4 , because of 0 g int dom f. Since dom f is open, l x x f dom f and then f Ž l x x . s q`. But, for any l g w0, l x ., we have always f Ž l x . y f Ž0. F pŽ l x .. Letting l ª l x , it leads to pŽ l x x . s l x pŽ x . s q`. Hence, pŽ x . s q`. COROLLARY 4.1. If wU -cl cow f Ž E .x has the RNP, then e¨ ery proper l.s.c. con¨ ex function g on E with g F f has the GFD. COROLLARY 4.2. A Banach space is an Asplund space if and only if there exists a proper l.s.c. con¨ ex function f such that f has an open and bounded effecti¨ e domain dom f and that e¨ ery proper l.s.c. con¨ ex function g on E with g F f has the GFD. Proof. If E is an Asplund space, then f Ž x. s
½
y1r Ž 5 x 5 y 1 . , q`
if 5 x 5 - 1, if 5 x 5 G 1,
is a required convex function. Conversely, if there exists such a convex function, then by Proposition 4.1, pŽ x . becomes d04 and by Theorem 3.1, pŽ0. s wU -cl cow f Ž E .x s EU has the RNP. THEOREM 4.1. Let f be a proper l.s.c. con¨ ex function on E with an open dom f. Then e¨ ery proper l.s.c. con¨ ex function g on E with g F f has the GFD if and only if wU -cl cow f Ž E .x has the RNP. Proof. We need only to show the necessity. Suppose that every continuous convex function g on E with g F f has the GFD, but CU s wUcl cow f Ž E .x has no RNP. Then we define that C1U [ wU-cl co Ž xU , yf U Ž xU . . g EU = R: xU g f Ž E . 4 . It is easy to see that C1U has no RNP in EU = R. Hence, there is a bounded non-empty subset AU1 ; C1U such that it does not admit wU-slices of arbitrarily small diameter and by Proposition 2.3, sAU1 : E = R ª R, which is dominated by sC 1U , is nowhere Frechet differentiable. ´ By definition and noting that we assume that f Ž0. s y1, we have that for any xU g f Ž x ., yf U Ž xU . s f Ž x . y ² xU , x : s inf
ygE
f Ž y . y ² xU , y : 4 F f Ž 0 . s y1.
Hence, for any x g E and t G 0,
sAU1 Ž x, t . F
U
sup
x g f Ž E .
² xU , x : y tf U Ž xU . 4 F f Ž x . ,
and t ¬ sAU1 Ž x, t . is non-increasing.
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LIXIN, SHUZHONG, AND LEE
The following proof is similar to that of Proposition 6.5 of w6x. Choose a differentiable non-positive function g: Ž1, 2. ª Ry such that g Ž3r2. s 0 and lim t ª 1 g Ž t . s lim t ª 2 g Ž t . y `. We define h: E ª R j q`4 by ; x g E,
h Ž x . [ sup
sA
U 1
Ž x, t . q g Ž t . 4 .
tg Ž1, 2 .
Then, h is l.s.c. convex and ; x g E,
h Ž x . F sup sAU1 Ž x, t . F sAU1 Ž x, 1 . F f Ž x . . tg Ž1, 2 .
Therefore, h has the GFD and there exists some point x 1 g int dom h such that h is Frechet differentiable at x 1. It is easy to check that there ´ exists t 1 g Ž1, 2. such that hŽ x 1 . s sAU1 Ž x 1 , t 1 . q g Ž t 1 .. Thus, by the Frechet ´ differentiability of h at x 1 and g at t 1 , we have that for any « ) 0, there exists d ) 0 such that g Ž t1 " t . q g Ž t1 . t . y 2 g Ž t1 . F t « and 0 F h Ž x 1 q ty . q h Ž x 1 y ty . y 2 h Ž x 1 . F t « whenever 5 y 5 s 1 and t g Ž0, d .. It leads to 0 F sAU1 Ž x 1 q ty, t 1 " t . q sAU1 Ž x 1 y ty, t 1 . t . y 2 sAU1 Ž x 1 , t 1 . F h Ž x 1 q ty . q h Ž x 1 y ty . y 2 h Ž x 1 . y g Ž t1 " t . q g Ž t1 . t . y 2 g Ž t1 . F t« , which contradicts that sAU1 is nowhere Frechet differentiable in E = R. ´ 5. PROOF OF THE MAIN THEOREM Now we prove our Main Theorem. Obviously, by Theorem 4.1, we have that Ži. m Živ. « Žiii. « Žii.. So, we need only show that Žii. implies Živ.. Suppose that f Ž E . is separable on each separable subspace of E. Without loss of generality, we still assume that f Ž0. s y1. Let p be the Minkowski functional generated by epi f. By Propositions 2.2 and 4.1, it is enough to show that pŽ0, 0. has the RNP; i.e., pŽ0, 0. is separable on each separable subspace E = R. By Proposition 2.1, it is equivalent to showing pŽgraph f . is separable on each separable subspace E = R. But
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by Proposition 2.2, the separability of f Ž E . on each separable subspace of E implies that pŽgraph f . is separable on each separable subspace of E = R.
6. CONCLUDING REMARKS In our Main Theorem, if f is the norm of E, then, after some simple deduction, we can also return to the classic result: E is an Asplund space if and only if EU has the RNP. It signifies that the nature of an Asplund space is the character of the norm as a dominant of a class of generically Frechet differentiable convex functions. ´ The condition of the openness of dom f in the Main Theorem is only so that graph f would be the boundary of epi f. This restriction is not serious. We will remove it in another paper.
REFERENCES 1. E. Asplund, Frechet differentiability of convex functions, Acta Math. 121 Ž1968., 31]47. ´ 2. R. D. Bourgin, Geometric aspects of convex sets with Radon]Nikodym ´ property, in ‘‘Lecture Notes in Math.,’’ Vol. 993, Springer-Verlag, New YorkrBerlin, 1983. 3. J. R. Giles and S. Sciffer, Separable determination of Frechet differentiability of convex ´ functions, Bull. Austral. Math. Soc. 52 Ž1995., 161]167. 4. P. S. Kenderov, Monotone operators in Asplund spaces, C. R. Acad. Bulgare Sci. 30 Ž1977., 963]964. 5. I. Namioka and R. R. Phelps, Banach spaces which are Asplund spaces, Duke Math. J. 42 Ž1975., 735]750. 6. R. R. Phelps, Convex functions, differentiability and monotone operators, in ‘‘Lecture Notes in Math.,’’ Vol. 364, Springer-Verlag, New YorkrBerlin, 1989. 7. Tang Wee-Kee, On Frechet differentiability of convex functions on Banach spaces, ´ Comment. Math. Uni¨ . Carolin. 36, No. 2 Ž1995., 249]253.
214, 367]377 Ž1997.
AY975570
Generic Frechet Differentiability of Convex Functions ´ on Non-Asplund Spaces Cheng Lixin and Shi Shuzhong Nankai Institute of Mathematics, Nankai Uni¨ ersity, Tianjin, 300071, People’s Republic of China
and E. S. Lee Department of Industrial Engineering, Kansas State Uni¨ ersity, Manhattan, Kansas 66506-5101 Submitted by William F. Ames Received April 4, 1997
Let f be a continuous convex function on a Banach space E. This paper shows that every proper convex function g on E with g F f is generically Frechet ´ differentiable if and only if the image of the subdifferential map f of f has the Radon]Nikodym ´ property, and in this case it is equivalent to showing that the image of f is separable on each separable subspace of E. Q 1997 Academic Press
1. INTRODUCTION Starting with Asplund’s work w1x, the notion of an Asplund space first appeared in Namioka and Phelps w5x: A Banach space is said to be an Asplund space if every Žreal-valued. continuous convex function defined on a non-empty convex open subset of the space is Frechet differentiable on a ´ dense Gd subset of its domain. Now, it is well known that a Banach space E is an Asplund space if and only if its dual EU has the Radon]Nikodym ´ property Žsee, for instance, w6x.. However, on a non-Asplund space, one can still find many non-trivial continuous convex functions, which are Frechet ´ differentiable on a dense Gd subset of its domain. 367 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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Recently, Tang w7x obtained the following theorem: THEOREM T. Suppose that f is a con¨ ex Lipschitz function on a separable Banach space E. Then the following statements are equi¨ alent: Ži. e¨ ery continuous con¨ ex function g on E with g F f is Frechet ´ differentiable on a dense Gd subset of E; Žii. the image of the subdifferential map f of f, f Ž E . s xU g EU : U x g f Ž x ., x g E4 is separable. Giles and Sciffer w3x showed that THEOREM GS. Let f be a continuous con¨ ex function on a Banach space E. If the image of the subdifferential map f of f is separable on each separable subspace of E, then f is Frechet differentiable on a dense Gd subset ´ of E. Both theorems show that on a separable Banach space E, which is not necessarily an Asplund space, and even on any Banach space E, there exist many generically Frechet differentiable continuous convex functions. ´ In this paper we will discuss similar problems and generalize and unify both theorems on any Banach space. The final result is as follows. MAIN THEOREM. Let f be a proper lower semi-continuous Ž l.s.c.. con¨ ex function on a Banach space E and its effecti¨ e domain dom f be open. Then the following statements are equi¨ alent: Ži. e¨ ery proper l.s.c. con¨ ex function g on E with g F f is Frechet ´ differentiable on a dense Gd subset of its effecti¨ e domain dom g; Žii. the image of the subdifferential map f of f, f Ž E . s xU g EU : U x g f Ž x ., x g E4 , is separable on each separable subspace of E; Žiii. the image of the subdifferential map f of f, f Ž E ., has the Radon]Nikodym ´ property; Živ. the wU -closed con¨ ex hull of the image of the subdifferential map f of f, wU -cl cow f Ž E .x, has the Radon]Nikodym ´ property. Taking f to be the indicator function of 04 , this theorem also generalizes the classic result: A Banach space E is an Asplund space if and only if its dual EU has the Radon]Nikodym ´ property or EU is separable on each separable subspace of E. The paper is organized as follows. In the next section, we introduce the notation and recall some preliminaries. Sections 3 and 4 discuss the generic Frechet differentiability of convex functions dominated by a lower ´ semicontinuous sublinear function or a continuous convex function. Section 5 completes the proof of the Main Theorem. In the last section, some concluding remarks are made.
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2. NOTATION AND PRELIMINARIES We will always denote a Banach space by E and its dual by EU . ² ? , ? : is the dual product on EU = E. Let f : E ª R j q`4 be a proper extended U real-valued function on E. Then the subdifferential map f : E ª 2 E of f is defined by
f Ž x . s xU g EU : ; y g E, f Ž y . y f Ž x . G ² xU , y y x : 4 . The conjugate function f U : EU ª R j q`4 of f is defined by f U Ž xU . s sup ² xU , x : y f Ž x . 4 . xgE
It is easy to check that
f Ž x . s xU g EU : f Ž x . q f U Ž xU . s ² xU , x : 4 . When f is convex on E and continuous at x, then f Ž x . is non-empty. If f is lower semi-continuous Žl.s.c.. and convex, i.e., its epigraph, defined by epi f [ Ž x, a . g E = R: f Ž x . F a 4 , U
is closed and convex, then f : E ª 2 E is a maximal monotone map w6, p. 59x; and ; x g E,
f Ž x . s sup
xUgE U
² xU , x : y f U Ž xU . 4 .
In addition, in this case, f is continuous on the interior of its effective domain, int dom f, where the effective domain of f is defined by dom f [ x g E: f Ž x . - q` 4 . See w6, p. 41x. In particular, if dom f is open, then f is continuous in dom f and the graph of f, graph f [ Ž x, f Ž x . . : x g dom f 4 , becomes the boundary of epi f. For a convex function f on E, f is Frechet differentiable at x g E if and ´ only if for any « ) 0 there exists d ) 0 such that 0 F f Ž x q ty . q f Ž x y ty . y 2 f Ž x . - t « whenever 5 y 5 s 1 and t g Ž0, d .. It is also equivalent to showing that f is single-valued and norm-to-norm upper semi-continuous at x w6, p. 48x.
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Every l.s.c. sublinear function p: E ª R j q`4 is the support function of a subset of EU ; i.e., there exists a subset CU ; EU such that ; x g E,
p Ž x . s sC U Ž x . [ sup ² xU , x : . xUgC U
p is continuous on E if and only if CU is bounded. It is easy to show that we can take CU s p Ž 0 . s xU g EU : ; y g E, ² xU , y : F p Ž y . 4 , and thus pU Ž xU . s dC U Ž xU ., the indicator function of CU , which is valued 0 on CU and q` out CU . Hence, when x / 0, xU g p Ž x . m xU g C U
and
² xU , x : s p Ž x . .
A non-negative continuous sublinear function on E is called a Minkowski functional. Every Minkowski functional pC : E ª Rq associates with a unique closed convex set C ; E with 0 g int C such that ; x g E,
pC Ž x . s inf l ) 0: x g lC 4 ,
and C s x g E: p Ž x . F 1 4 . We also say that pC is the Minkowski functional generated by C. In this case, CU is the polar set of C; i.e., CU s xU g EU : ; x g C, ² xU , x : F 1 4 . Let S [ C s x g E: pC Ž x . s 14 . Then pC Ž S . consists of all support functionals xU of C at x with ² xU , x : s 1. By the Bishops]Phelps theorem w6, pp. 51]52x, the support functionals are dense in the cone of all those functionals which are bounded above on C. In particular, we have that PROPOSITION 2.1. pC Ž S . is dense in C1U [ xU g EU : sup x g C ² xU , x : s 14 and thus, pC Ž0. s cl co pC Ž S . j 044 . We will use the Minkowski functional generated by the epigraph of a l.s.c. convex function with an open dom f. In this case, we have that C s epi f and S s graph f. We need the following proposition: PROPOSITION 2.2. Let f : E ª R j q`4 be a proper l.s.c. con¨ ex function on E and continuous at x s 0 with f Ž0. s y1. Let p: E = R ª R be
371
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the Minkowski functional generated by epi f. Then xU g f Ž x . if and only if
Ž yU , r . g p Ž x, f Ž x . .
with yU s xUrf U Ž xU .
and r s y1rf U Ž xU . ; f is Frechet differentiable at x g E if and only if p is Frechet differentiable at ´ ´ Ž x, f Ž x .. g E = R. Proof. In fact, xU g f Ž x . is equivalent to f Ž x . q f U Ž xU . s ² xU , x :; i.e., 1 U
f Ž xU .
² xU , x : y f Ž x . s p Ž x, f Ž x . . s 1,
and noting that ; Ž y, r . g epi f ,
² xU , x : y f Ž x . G ² xU , y : y f Ž y .
s² Ž xU , y1 . , Ž y, f Ž y .: G² Ž xU , y1 . , Ž y, r .: , we have that Ž xU , y1.rf U Ž xU . g pŽ x, f Ž x ... Here, we recall that f Ž0. s y1 implies that f U Ž xU . s sup ² xU , x : y f Ž x . 4 G yf Ž 0 . s 1. xgE
From this equivalent relation, it is easy to check that f is single-valued and norm-to-norm upper semi-continuous at x if and only if p with
p Ž x, f Ž x . . s Ž yU , r . g EU = R: yU s
½
xU U
² x , x: y f Ž x .
,rs
y1 U
² x , x: y f Ž x .
5
is single-valued and norm-to-norm upper semi-continuous at Ž x, f Ž x ... Let AU ; EU , x g E with x / 0 in E and a ) 0. Then we say that S Ž x, AU , a . [ xU g AU : ² xU , x : ) sAU Ž x . y a 4 is a wU -slice of AU , where sAU Ž x . [ sup xU g AU Ž x . is the support function of AU . We say that a non-empty subset AU of EU is wU-dentable, if it admits wU-slices of arbitrarily small diameter, i.e., for every « ) 0, there exist x g E with x / 0 and a ) 0 such that diam S Ž x, AU , a . - « .
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A subset AU ; EU is said to have the Radon]Nikodym ´ property ŽRNP. if Ži. every non-empty bounded subset of AU is wU -dentable. If AU is wU-closed and convex, then AU having the RNP is equivalent to saying that Žii. AU is separable on each separable subspace of E; i.e., for every separable subspace E1 of E, AU < E1 is separable. ŽSee w2, p. 91x for more equivalent conditions.. In particular, EU is said to have the RNP if every separable subspace of E has a separable dual. The following theorem is due to Kenderov Žw4x; see also w6, p. 32x; the proof was only for CU s EU , but it is suitable for any CU ; EU .: THEOREM 2.1. Assume that CU ; EU is wU-closed con ¨ ex and has the U RNP. Then for e¨ ery maximal monotone map T : E ª 2 C with int DŽT . / B, where DŽT . [ x g E: T Ž x . / B4 , there is a dense Gd subset G of DŽT . such that T is single-¨ alued and norm-to-norm upper semi-continuous at each point of G. A proper l.s.c. convex function f on E with int dom f / B is said to be generically Frechet differentiable on dom f if f is Frechet differentiable on a ´ ´ dense Gd subset G of dom f. We will say that this function has the Generic Frechet Differentiability ŽGFD.. By Theorem 2.1, f has the GFD if the ´ U w -closed convex hull of f Ž E ., denoted by wU -cl cow f Ž E .x, has the RNP, because f is maximal monotone. The proof of the following proposition can be found in w6, pp. 25]26x. PROPOSITION 2.3. Suppose that AU ; EU is bounded and does not admit w -slices of arbitrarily small diameter. Then the continuous support function sAU is nowhere Frechet differentiable. ´ U
3. GFD OF CONVEX FUNCTIONS DOMINATED BY A L.S.C. SUBLINEAR FUNCTION In this section, we will seek a necessary and sufficient condition for the GFD of convex functions dominated by a l.s.c. sublinear function. THEOREM 3.1. Suppose that p: E ª R j q`4 is a l.s.c. sublinear function. Then the following statements are equi¨ alent: Ži. Žii. Žiii.
e¨ ery proper l.s.c. con¨ ex function g with g F p has the GFD; e¨ ery l.s.c. sublinear function g with g F p has the GFD; pŽ0. has the RNP.
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If p is a Minkowski functional, then they are also equi¨ alent to Živ. e¨ ery Minkowski functional g with g F p has the GFD; Žv. for S s x g E: pŽ x . s 14 , pŽ S . is separable on each separable subspace of E. Proof. Let CU [ pŽ0. s xU g X U : ; x g E, ² xU , x : F pŽ x .4 . Then Ž p x . s sC U Ž x .. Suppose that Ži. holds; i.e., every proper l.s.c. convex function g with g F p has the GFD. Then Žii. and Živ. hold. If CU has no RNP, then there exists a bounded subset AU ; CU such that AU does not admit wU-slices of arbitrarily small diameter. Let sAU be the support function of AU . Then sAU is a continuous convex function on E. By definition, we have that sAU F sC U s p, and by Proposition 2.3, sAU is nowhere Frechet differentiable, which leads to a contradiction. Hence, Žii. ´ Ž . implies iii . In the case that p is a Minkowski functional, we can take g 1 [ sup sAU , 04 . Then g 1 is also a Minkowski functional. But it has no the GDF. This contradiction shows that Živ. implies Žiii.. Conversely, if Žiii. holds, i.e., CU has the RNP, then we can show that for every l.s.c. convex function g with g F p, the image of g is in CU , i.e., U g: E ª 2 C and then, by Theorem 2.1, g has the GFD. In fact, if there exists xU g g Ž x . with xU f CU , then by the Separation Theorem, there exists z g E and d ) 0 such that ² xU , z : ) d q sC U Ž z . s d q p Ž z . , which implies z g dom p. Since xU g g Ž x ., we have that ; y g E, g Ž y . y g Ž x . G ² xU , y y x : . Hence, taking y s kz, we obtain that g Ž kz . y g Ž x . G ² xU , kz : y ² xU , x : ) k d q p Ž kz . y ² xU , x : , which implies g Ž kz . ) pŽ kz . for all sufficiently large k ) 0 and leads to a contradiction with the definition of g. Thus, Žiii. implies Ži.. Finally, if p is a Minkowski functional, then, by Proposition 2.1, it is easy to check that Žiii. is equivalent to Žv.. COROLLARY 3.1.
E is an Asplund space if and only if EU has the RNP.
In this case, we can take CU s EU . Then pC U s d04 and pŽ0. s EU . Theorem 3.1 may be modified for convex functions ‘‘locally’’ dominated by a l.s.c. sublinear function. THEOREM 3.2. Suppose that p: E ª R j q`4 is a l.s.c. sublinear function. Gp is the collection of all l.s.c. con¨ ex functions g satisfying that for any x g dom g, there exists d x ) 0 and c x ) 0 such that ; y g dom g with 5 y y x 5 - d x , g Ž y . y g Ž x . F cx pŽ y y x . . Then e¨ ery g g Gp has the GFD if and only if pŽ0. has the RNP.
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Proof. Let CU [ pŽ0.. Then pŽ x . s sC U Ž x .. Suppose that every g g Gp has the GFD. If CU has no RNP, then there exists a bounded subset AU ; CU such that AU does not admit wU-slices of arbitrarily small diameter. Let sAU be the support function of AU . Then sAU is a continuous convex function on E. By definition, we have that sAU F sC U s p, and then, by the sublinearity of sAU , ; x, y g E,
sAU Ž y . y sAU Ž x . F sAU Ž y y x . F sC U Ž y y x . .
Hence, sAU g Gp . Again by Proposition 2.2, sAU is nowhere Frechet ´ differentiable, which leads to a contradiction. Conversely, it is easy to check that for any g g Gp , g Ž E . ; CNU [ D ng N n pŽ0.. Since 0 g pŽ0., pŽ0. has the RNP if and only if CNU has the RNP. Again by Theorem 2.1, g has the GFD. COROLLARY 3.2. Assume that p is a Frechet differentiable Ž not necessarily ´ . equi¨ alent norm. Then e¨ ery p-continuous con¨ ex function has the GFD. This is because we can check that pŽ0. has the RNP and that every p-continuous convex function is locally p-Lipschitz as in the case that p is the proper norm of E.
4. GFD OF CONVEX FUNCTIONS DOMINATED BY A CONTINUOUS CONVEX FUNCTION Theorem 3.1 may hold in the case where a l.s.c. sublinear function is replaced by a proper l.s.c. convex function with an open effective domain. This convex function is continuous on its domain. Let f be a proper l.s.c. convex function on E with an open dom f, and CU s f Ž E .. Then for pŽ x . s sup xU g C U ² xU , x :, we have that pŽ0. s wU -cl cow f Ž E .x. Without loss of generality, we will always assume that 0 g int dom f and f Ž0. s y1. PROPOSITION 4.1. Under the abo¨ e assumption, the following inequality holds: ; x g E,
f Ž x . y f Ž 0. F p Ž x . .
Proof. If x s 0, the inequality is trivial. If x / 0 and x g dom f, then f is continuous at x and f Ž x . is non-empty. Thus, for xU g f Ž x ., f Ž x . y f Ž 0 . F ² xU , x : F sup ² xU , x : s p Ž x . . xUgC U
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If x f dom f, i.e., f Ž x . s q`, then we show that pŽ x . s q`. In fact, in this case, there exists l x g Ž0, 1. such that l x s sup l g Ž0, 1.: l x g dom f 4 , because of 0 g int dom f. Since dom f is open, l x x f dom f and then f Ž l x x . s q`. But, for any l g w0, l x ., we have always f Ž l x . y f Ž0. F pŽ l x .. Letting l ª l x , it leads to pŽ l x x . s l x pŽ x . s q`. Hence, pŽ x . s q`. COROLLARY 4.1. If wU -cl cow f Ž E .x has the RNP, then e¨ ery proper l.s.c. con¨ ex function g on E with g F f has the GFD. COROLLARY 4.2. A Banach space is an Asplund space if and only if there exists a proper l.s.c. con¨ ex function f such that f has an open and bounded effecti¨ e domain dom f and that e¨ ery proper l.s.c. con¨ ex function g on E with g F f has the GFD. Proof. If E is an Asplund space, then f Ž x. s
½
y1r Ž 5 x 5 y 1 . , q`
if 5 x 5 - 1, if 5 x 5 G 1,
is a required convex function. Conversely, if there exists such a convex function, then by Proposition 4.1, pŽ x . becomes d04 and by Theorem 3.1, pŽ0. s wU -cl cow f Ž E .x s EU has the RNP. THEOREM 4.1. Let f be a proper l.s.c. con¨ ex function on E with an open dom f. Then e¨ ery proper l.s.c. con¨ ex function g on E with g F f has the GFD if and only if wU -cl cow f Ž E .x has the RNP. Proof. We need only to show the necessity. Suppose that every continuous convex function g on E with g F f has the GFD, but CU s wUcl cow f Ž E .x has no RNP. Then we define that C1U [ wU-cl co Ž xU , yf U Ž xU . . g EU = R: xU g f Ž E . 4 . It is easy to see that C1U has no RNP in EU = R. Hence, there is a bounded non-empty subset AU1 ; C1U such that it does not admit wU-slices of arbitrarily small diameter and by Proposition 2.3, sAU1 : E = R ª R, which is dominated by sC 1U , is nowhere Frechet differentiable. ´ By definition and noting that we assume that f Ž0. s y1, we have that for any xU g f Ž x ., yf U Ž xU . s f Ž x . y ² xU , x : s inf
ygE
f Ž y . y ² xU , y : 4 F f Ž 0 . s y1.
Hence, for any x g E and t G 0,
sAU1 Ž x, t . F
U
sup
x g f Ž E .
² xU , x : y tf U Ž xU . 4 F f Ž x . ,
and t ¬ sAU1 Ž x, t . is non-increasing.
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The following proof is similar to that of Proposition 6.5 of w6x. Choose a differentiable non-positive function g: Ž1, 2. ª Ry such that g Ž3r2. s 0 and lim t ª 1 g Ž t . s lim t ª 2 g Ž t . y `. We define h: E ª R j q`4 by ; x g E,
h Ž x . [ sup
sA
U 1
Ž x, t . q g Ž t . 4 .
tg Ž1, 2 .
Then, h is l.s.c. convex and ; x g E,
h Ž x . F sup sAU1 Ž x, t . F sAU1 Ž x, 1 . F f Ž x . . tg Ž1, 2 .
Therefore, h has the GFD and there exists some point x 1 g int dom h such that h is Frechet differentiable at x 1. It is easy to check that there ´ exists t 1 g Ž1, 2. such that hŽ x 1 . s sAU1 Ž x 1 , t 1 . q g Ž t 1 .. Thus, by the Frechet ´ differentiability of h at x 1 and g at t 1 , we have that for any « ) 0, there exists d ) 0 such that g Ž t1 " t . q g Ž t1 . t . y 2 g Ž t1 . F t « and 0 F h Ž x 1 q ty . q h Ž x 1 y ty . y 2 h Ž x 1 . F t « whenever 5 y 5 s 1 and t g Ž0, d .. It leads to 0 F sAU1 Ž x 1 q ty, t 1 " t . q sAU1 Ž x 1 y ty, t 1 . t . y 2 sAU1 Ž x 1 , t 1 . F h Ž x 1 q ty . q h Ž x 1 y ty . y 2 h Ž x 1 . y g Ž t1 " t . q g Ž t1 . t . y 2 g Ž t1 . F t« , which contradicts that sAU1 is nowhere Frechet differentiable in E = R. ´ 5. PROOF OF THE MAIN THEOREM Now we prove our Main Theorem. Obviously, by Theorem 4.1, we have that Ži. m Živ. « Žiii. « Žii.. So, we need only show that Žii. implies Živ.. Suppose that f Ž E . is separable on each separable subspace of E. Without loss of generality, we still assume that f Ž0. s y1. Let p be the Minkowski functional generated by epi f. By Propositions 2.2 and 4.1, it is enough to show that pŽ0, 0. has the RNP; i.e., pŽ0, 0. is separable on each separable subspace E = R. By Proposition 2.1, it is equivalent to showing pŽgraph f . is separable on each separable subspace E = R. But
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by Proposition 2.2, the separability of f Ž E . on each separable subspace of E implies that pŽgraph f . is separable on each separable subspace of E = R.
6. CONCLUDING REMARKS In our Main Theorem, if f is the norm of E, then, after some simple deduction, we can also return to the classic result: E is an Asplund space if and only if EU has the RNP. It signifies that the nature of an Asplund space is the character of the norm as a dominant of a class of generically Frechet differentiable convex functions. ´ The condition of the openness of dom f in the Main Theorem is only so that graph f would be the boundary of epi f. This restriction is not serious. We will remove it in another paper.
REFERENCES 1. E. Asplund, Frechet differentiability of convex functions, Acta Math. 121 Ž1968., 31]47. ´ 2. R. D. Bourgin, Geometric aspects of convex sets with Radon]Nikodym ´ property, in ‘‘Lecture Notes in Math.,’’ Vol. 993, Springer-Verlag, New YorkrBerlin, 1983. 3. J. R. Giles and S. Sciffer, Separable determination of Frechet differentiability of convex ´ functions, Bull. Austral. Math. Soc. 52 Ž1995., 161]167. 4. P. S. Kenderov, Monotone operators in Asplund spaces, C. R. Acad. Bulgare Sci. 30 Ž1977., 963]964. 5. I. Namioka and R. R. Phelps, Banach spaces which are Asplund spaces, Duke Math. J. 42 Ž1975., 735]750. 6. R. R. Phelps, Convex functions, differentiability and monotone operators, in ‘‘Lecture Notes in Math.,’’ Vol. 364, Springer-Verlag, New YorkrBerlin, 1989. 7. Tang Wee-Kee, On Frechet differentiability of convex functions on Banach spaces, ´ Comment. Math. Uni¨ . Carolin. 36, No. 2 Ž1995., 249]253.