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Acta Mathematica Vietnamica

ON THE SUBDIFFEENTIAL OF AN UPPER ENVELOPE OF CONVEX FUNCTIONS

icon-email M. VOLLE

Abstract

We extend a Rockefellers result for the subdifferential of the upper envelope j=sup1inf, of a finite collection f1,,fr of convex proper functionals on a locally convex Hausdorff topological space X. Assuming that f1,,fn1 are finite and continuous at a point x0 of X where fn is finite, we show that, for any point x of X such that f(x) is finite (*)f(x)=co{fk(x):fk(x)=f(x)}+i=1nN(dom fi,x), where co stands for the convex hull and N(dom fi,x) for the normal cone to the domain dom fi of fi at x. We also give an application of (*) to asymptotical analysis related to a result by Choquet, and prove that (*) remains true when the epigraph of the Legendre-Fenchel conjugate of f is weak* complete and pointed, and the fi are lower-semicontinuous.