0 < f(y) - fix) - (x*, y - x) < e\\y - x\\

Transcription

1 proceedings of the american mathematical society Volume 121, Number 4, August 1994 A NOTE ON THE DIFFERENTIABILITY OF CONVEX FUNCTIONS WU CONGXIN AND CHENG LIXIN (Communicated by Palle E. T. Jorgensen) Abstract. Every real-valued convex and locally Lipschitzian function / defined on a nonempty closed convex set D of a Banach space E is the local restriction of a convex Lipschitzian function defined on E. Moreover, if E is separable and int D ± 0, then, for each Gateaux differentiability point x (6 int/j) of f, there is a closed convex set C C intd with the nonsupport points set N(C) ^ 0 and with x e N(C) such that fc (the restriction of / on C) is Fréchet differentiable at x. Rainwater [1] and Verona [2] recently generalized Asplund's theorems [5] in the following manner Theorem [Rainwater]. Suppose that E is an Asplund space (a Banach space of class (S)). Then, for every closed convex subset C of E such that the nonsupport points set NÍC) of C is nonempty and for every convex function f on C which is locally Lipschitzian on NÍC), the set of points Q, where f is Fréchet ÍGateaux) differentiable, is a dense G subset of C. Notation. We denote by C, Cx, and N(C) a closed convex set of a Banach space E, the cone generated by C from x, and the nonsupport points set of C, respectively. The open and the closed balls centered at x and with radius r are denoted by 5(x, r) and 5(x, r), respectively. All convex functions /, if nothing is added, are assumed real valued. Definition. We say that a function denoted by / is a local extension of f at u dom(/) and that / is a local restriction of / at u provided there is a neighborhood U of u such that / = / in U n dom(/ ). Definition. We say that / is Gateaux (Fréchet) differentiable at x NÍC) provided there exists a unique x* E* such that (x\y-x)< fiy) - px) for all y G C (for all e > 0 there exists ô > 0 such that whenever y C n B(x, Ô)). 0 < f(y) - fix) - (x*, y - x) < e\\y - x\\ Received by the editors October 28, Mathematics Subject Classification. Primary 26E15, 46G American Mathematical Society /94 $1.00+ $.25 per page

2 1058 WU CONGXIN AND CHENG LIXIN Lemma 1. Suppose that f is convex and locally Lipschitzian on C. Then (a) for every u C, there exist a neighborhood U of u and a convex Lipschitzian function fu defined on E such that f = fu in U n C ; (b) dom(d/) = dom(/) = C; (c) there exists a selection y far df on C such that y/ is locally bounded on C ; in particular [1], if NÍC) # 0, then df is itself locally bounded on N(C). Recall that the extended real-valued lower semicontinuous convex function f on E is said to be proper provided fix) > -oo for all x E and its iconvex) essential domain domf = {x: fix) < +00} is nonempty. For such f, we define the inf-convolutions as fnix) = inf{fiy) + n\\x -y\\: y E}. Then we have isee, for instance, [3, 4, 6]). Lemma 2. With f and {/ } as above, the sequence {/ } has the following properties : Í1 ) Each fn is convex and Lipschitzian on E with Lipschitz constant n ; (2) f ix) < f +iix) < fix) far each x E and each n > 1 ; (3) fnix) = fix) if and only if d fix) HnB* {where B* denotes the unit ball of E*) is nonempty or, equivalently, if and only if dfn(x) = dfix) n nb*. Proof of Lemma 1. Since / is locally Lipschitzian on C, there exists B(u, r) for some r > 0 and L > 0 such that \f(y) - /C*) ^ -^11v ~ xll whenever x, y G B(u, r) n C = Bciu,r). We define the extended real-valued convex function / on E by fix) - fix) if x Bciu, r), and fix) - 00, otherwise. Hence / is proper lower semicontinuous convex and bounded below on E and Lipschitzian on 2?c(«,r) with Lipschitz constant L. Let {/ } be the sequence of the inf-convolutions by /. Now we will show (a). Suppose, to the contrary, that the neighborhood U of u does not exist. Let Bn = Bciu, 1/n). Then, for each integer n > 1, there is x Bn such that / (x ) < /(x ), by the definition of /. For each such xn, we can choose y E such that f(xn)>fiyn) + n\\yn-xn\\, n = 1,2,..., that is, fixn) - fiy ) > n\\y -xn\\ for n = 1, 2,.... Clearly, we have y Bc(u, r). Note that xn Bc(u, r) whenever 1/n < r and / is Lipschitzian on Bc(u, r) with Lipschitz constant L. We have \f(x ) - f(yn)\ <L\\x -y \\ whenever l/n < r ; this is a contradiction which proved assertion (a). From the proof of assertion (a), it is easy to see that fn = fin Bc(u, r) by taking n [L] + 1 (where [L] denotes the maximal integer m satisfying m < L) ; that is, by Lemma 2, df(x)r\nb* is nonempty for each x Bc(u, r). Hence df(x) n nb* is nonempty for each x ß(w, r). Now we showed that there is a selection for df on B(u,r) which is bounded by n. The arbitrariness of u says that there is a selection ^ for 9/ on C which is locally bounded; hence we proved assertions (b) and (c). With / and / as in Lemma 1, by Lemma 2, we have dfu(x) c df(x) for each x G U n C. If C is closed and N(C) 0, note that x N(C) if

3 A NOTE ON THE DIFFERENTIABILITY OF CONVEX FUNCTIONS 1059 and only if Cx is dense in E [1]. By a simply convexity and differentiability argument, we see df = dfu in U C\ N(C). Hence we have Proposition 3. Suppose that C is closed with N(C) ^ 0 and f is locally Lipschitzian on N(C). Then the following versions are equivalent: (a) / is Gateaux differentiable at u N(C). (b) Every fu is Gateaux differentiable at u N(C). (c) Every selection for df on N(C) is norm-to-weak* continuous at u. (d) There is a selection for df on N(C) which is norm-to-weak* continuous at u. Corollary 4. Suppose that C is a closed convex set of the weak Asplund space E, and suppose that the convex function f is locally Lipschitzian on N(C). Then the set of points G, where f is Gateaux differentiable, is a Gs set of C. It is easy to see that if u N(C) is the Fréchet differentiability point of fu, then so is the one of f. The following example shows that if the interior of C is empty, then the Fréchet differentiability of fu is really stronger than the one off. ' Example 5. The function / defined by f(x) - \\x\\p on C = {x ll, x > 0 for n = 1,2,...} shows that / is Fréchet differentiable at each point of N(C) {x lx, xn > 0 for «= 1,2,...}, but the extension / of / (where fu II II/' on /') is nowhere Fréchet differentiable. More generally, we have Theorem 6. Suppose that E is separable, D is a nonempty open convex set of E, and f is a continuous convex function on D. Then, for each Gateaux differentiability point x D of f, there exists a closed convex subset C of D with N(C) t 0 and x N(C) such that fc (the restriction of f to C) is Fréchet differentiable at x. Proof. Let 0 < e < \. By [8] there are two sequences {xf} and {Xj} in E and in E*, respectively, satisfying (1) ^ = 1 for 7 = 1,2,...; (2) x; <l+e for; = 1,2,...; (3) x*(xj) = Sij (=1 if 1=/, =0 otherwise); (4) cl(span(x, for j = 1, 2,...)) = E. Suppose that x D is a Gateaux differentiability point of /. each n > 1, there exists 1 > 5n > 0 (5n > 0) such that (1) f(x±2ô xn)-fix)-(x*,±2ônxn) <2_n {x. = df{x)) Then, for Let En = span(x, for j = 1, 2,...,«), F = cë(±y for n = 1,2,...), and C = x + F, where y = ô x for n = 1,2,... Clearly, C (c D) is closed and convex and Cx is dense in E, and hence xéjv(c). Now we prove that point x is a Fréchet differentiability point of f. Suppose, to the contrary, that fc is not Fréchet differentiable at x. Then there is en > 0 such that for each

4 1060 WU CONGXIN AND CHENG LIXIN Sn there is z C with 0 < \\z - x\\ < dn such that fc(zn)-fc(x)-(x*,z -x) ^ ( - >*> y where x* = dfix). By the density of coi±yn for n = 1, 2,... ) in F, we can assume that {z - x} c coi±y for n = 1,2,...), that is, for each n > 1, there is a sequence {k{"]} with k{"] > 0 for i = 1, 2,... and, JL[.B) = 1 such that zn -x = ^/^ "'(iy/) Therefore, we have, by the properties of {x } and llz«- *ll > 7 max x*(z - x)\ (2), ;, = -^ max k{"]\\yj\\ = maxä,. l+e j J " ;" l+e j J ' Fix «o such that (1 + e) ~> = (1 + e)2-*> < e0/2. Let «o oo un=x + 2Y/l,n)(±yi) and vn = x + 2 X?\±yi). 1=1 i'=no + l Since z x, we can claim m,ü ë D for n = 1,2. Now we have z = ("«+ "" )/2, therefore, /c(z ) = /(z ) = /((«+ vn)/2) < \[fiun) + f(vn)]. Let w (u - x)/6 (where 6 - \\z - x\\). Then IKH = \\u -x\\/e <YlMln)\\yi\\/0n = 2Yix<?)âi/e (3),=1 '=' < il+e)-2y/^l")ài/maxxf)ôj < 2(1 +e)n0; that is, {wn} is contained by 2(1 + e)nob 0, where B 0 is the unit ball of the «o-dimensional subspace E 0 of E. (oo \ / oo ftp x + 2 Xf(±yi)\=fi k\"\x±2yl) + Y/k^x (=«0+1 / \/'=/io+l 1=1 oo «o < *.(i")f(x±2yi) + y ix?).f(x). i'=no+l '=1

5 A NOTE ON THE DIFFERENTIABILITY OF CONVEX FUNCTIONS 1061 Hence o < f(zn) - fix) - (x*, zn -x) < {[fjun) + fjvn)]/2 - fjx) - (X*, Z - x)} e f(u ) - f(x) - (X*,Un-x) f(v ) - f(x) - (X*,Vn-x) n < f(un) - fjx) - (X*, Un-X) 26n + ZT=n0+i^](f(x± 2yt) - f(x) - (x*, ±2y,» 26 (2) f(u ) - f(x) - (X*, U -X) 2dn (1 + e) T,T=n0+i in)(f(x ± 2v,) - fix) - (x*, ±2y,)) 2maxjXfôj (xjf(un)-f(x)-(x*,u -x) (1 c) 2_j j=n0+l f(u )-f(x)-(x*,u -x) q 26n + 2 = fix + enwn) - f(x) - (x*, enwn) " Since 2(1 + e)«o^/io is a bounded set of the «o-dimensional subspace E 0 of, u) ë2(1+ s)nob 0, and ö -» 0, and since / is Gateaux differentiable at x, we must have fjx + dnw ) - fjx) - (x*, d w ) for sufficiently large n. Hence fjx + enwn)- fjx)- (X*, d W ), 0 «0, «0 e - 2^-+ T< 2" + T = e for sufficiently large «, and this is a contradiction which completes our proof. Remark. The recent work of Preiss, Phelps, and Namioka [7] showed that if E admits an equivalent smooth norm, then it is of class (S). Hence, for the Gateaux differentiability, the Rainwater theorem still holds for smoothable Banach spaces. Acknowledgment The second author specially thanks Professors R. R. Phelps and I. Namioka for their hospitality and helpful conversations on this note.

6 1062 WU CONGXIN AND CHENG LIXIN References 1. J. Rainwater, Yet more on the differentiability of convex functions, Proc. Amer. Math. Soc. 103(1988), M. E. Verona, More on the differentiability of convex functions, Proc. Amer. Math. Soc. 103 (1988), J. B. Hiriart Urruty, Lipschitz r-continuity of the approximate subdifferential of a convex junction, Math. Scand. 47 (1980), P. J. Laurent, Approximation et optimisation, Hermann, Paris, E. Asplund, Fréchet differentiability of convex functions, Acta. Math. 121 (1968), S. Fitzpatrick and R. R. Phelps, Bounded approximants to monotone operators on Banach spaces (to appear). 7. D. Preiss, R. R. Phelps, and I. Namioka, Smooth Banach spaces, weak Asplund spaces and monotone or USCO mappings (to appear). 8. J. Lindenstrauss and L. Tzafiri, Classical Banach spaces. I, Springer-Verlag, New York, Department of Mathematics, Harbin Institute of Technology, Harbin, , People's Republic of China