Linear Selections of Superlinear Set-Valued Maps with some Applications to Analysis

Abstract

A. Ya. Zaslavskii's results on the existence of a linear (affine) selection for a linear (affine) or superlinear (convex) map $\Phi : K \to 2^Y$ defined on a convex cone (convex set) $K$ having the interpolation property are extended. We prove that they hold true under more general conditions on the values of the mapping and study some other properties of the selections. This leads to a characterization of Choquet simplexes in terms of the existence of continuous affine selections for arbitrary continuous convex maps. A few applications to analysis are given, including a construction that leads to the existence of a (not necessarily bounded) solution for the corona problem in polydisk $\mathbb D^n$ with radial boundary values that are bounded almost everywhere on $\mathbb T^n$.