Convergence of best approximations in a smooth Banach space

Abstract

Let X be a reflexive, strictly convex Banach space such that both X and X∗ have Fréchet differentiable norms, and let {Cn} be a sequence of non-empty closed convex subsets of X. We prove that the sequence of best approximations {p(x ¦ Cn)} of any x ϵ X converges if and only if lim Cn exists and is not empty. We also discuss measurability of closed convex set valued functions.