On Gâteaux Differentiability of Convex Functions in WCG Spaces

It is shown, using the Borwein–Preiss variational principle that for every continuous convex function f on a weakly compactly generated space X, every x0 ∈ X and every weakly compact convex symmetric set K such that spanK = X, there is a point of Gâteaux differentiability of f in x0 +K. This extends a Klee’s result for separable spaces. The well-known Mazur’s theorem says that a continuous convex function f on a separable Banach space X is Gâteaux differentiable on a dense Gδ set, [4, Theorem 8.14]. A function f on X is said to be Gâteaux differentiable at x ∈ X if there is F ∈ X such that lim t→0 f (x + th)− f (x) t = F(h), for all h ∈ X. A Banach space is called a weak Asplund space if every continuous convex function f on it is Gâteaux differentiable at the points of a dense Gδ set. It is known that weakly compactly generated spaces are weak Asplund spaces, [3, Theorem 1.3.4]. Recall that a Banach space X is called weakly compactly generated (WCG) if there is a weakly compact set K ⊂ X such that spanK = X. It is proved in [5] that, for a separable Banach space X, the set of points of Gâteaux differentiability of a convex continuous function f is even bigger than dense in the following sense. If K ⊂ X is a norm compact convex symmetric set such that spanK = X and x0 ∈ X, then there is x ∈ x0 + K, a point of Gâteaux differentiability of f . A set C ⊂ X is called symmetric if −C = C . We will extend the above result to weakly compact set in WCG spaces. Theorem 1 Let X be a WCG space and K be a weakly compact convex symmetric set such that spanK = X. Let f be a continuous convex function on X and x0 ∈ X. Then there is x ∈ x0 + K such that f is Gâteaux differentiable at x. Let us define terms used in the proof. For a closed convex symmetric set C let μC denote a Minkowski functional ofC defined by μC(x) = inf{λ > 0 ; x ∈ λC}. Received by the editors October 21, 2003. Research supported by NSERC 7926, FS Chia Ph.D. Scholarship and Izaak Walton Killam Memorial Scholarship, written as part of Ph.D. thesis under supervision of Dr. N. Tomczak-Jaegermann and Dr. V. Zizler. AMS subject classification: 46B20.