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

TL;DR

This paper extends Zaslavskii's results on linear selections for superlinear set-valued maps, providing broader conditions, characterizations of Choquet simplexes, and applications including solutions to the corona problem in polydisks.

Contribution

It generalizes existing theorems on linear selections for superlinear maps and explores their properties, leading to new characterizations and applications in analysis.

Findings

Extended conditions for the existence of linear selections.

Characterization of Choquet simplexes via affine selections.

Constructed solutions to the corona problem in polydisks.

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 $\mathbb T^n$.