Gradients of convex functions
Edgar Asplund, R. TYRRELL ROCKAFELLAR · Transactions of the American Mathematical Society · 1969
Introduction.This paper is concerned with relationship between three notions: the differentiability of a convex function/ the rotundity of the convex function g conjugate to / and the continuity of the subdifferential mapping 8fi (which reduces to the gradient mapping V/where/is differentiable).These notions are considered in the context of various admissible topologies on paired vector spaces.When /is the norm || • || on a Banach space X, or/=(l/2)|| ■ ||2, our results are comparable to the theorems of Smulyan [19], Cudia [6] and others about the relationship between the differentiability of ¡|||, the rotundity of the dual unit ball in X* and the continuity of the spherical or extended spherical mappings from A" to A**; see Asplund [2].Our results also contain as a special case some recent results of Lescarret [9] on the strong continuity of gradient mappings in Banach spaces.They are stronger than, but do not quite contain, the theorems of Moreau [12], [13] about the upper semicontinuity of 8fi and SA/(see the remark following Proposition 5).We would like to thank Professor J. J. Moreau for some very helpful suggestions with regard to an earlier version of this paper.2. Basic definitions.Throughout this paper, X and F will denote vector spaces over the real number system F paired by a bilinear form , with respect to which X distinguishes the points of F and F distinguishes the points of X.We denote by w(X, Y) and s(X, Y), respectively, the weak and strong topologies induced on X by F; similarly w( Y, X) and s( Y, X) on F. Differentiability properties in the space X will be shown to be dual to rotundity properties in the space F.Let / be an extended-real-valued function on X (i.e. an everywhere-defined function with values in F u {±oo}).Let A be any nonempty subset of X.We shall say that/is A-differentiable at a given xe X if/is finite at x and there exists a y e Y such that f(^u)-fi(x)_<u^=0