On affine selections of set-valued functions

TL;DR

This paper proves that convex set-valued functions on real intervals with compact values in locally convex spaces always have an affine selection, and extends the result to certain non-convex cases in real line settings.

Contribution

It establishes the existence of affine selections for convex set-valued functions and generalizes the conditions under which such selections exist in real line cases.

Findings

Every convex set-valued function on a real interval admits an affine selection.

The convexity assumption can be relaxed in the real line case with closed intervals.

The results extend to functions with more general conditions in real line settings.

Abstract

The main result states that every convex set-valued function defined on a real interval with compact values in a locally convex space, admits an affine selection. In the case if the target space is a real line and the values are closed real intervals, we can replace the convexity assumption by the more general condition.