Existence Criteria for Lipschitz Selections of Set-Valued Mappings in ${\bf R}^2$

TL;DR

This paper develops constructive criteria and algorithms for determining the existence of Lipschitz selections for convex set-valued mappings in the plane, enabling nearly optimal solutions and Lipschitz seminorm estimates.

Contribution

It introduces new geometric criteria and efficient algorithms for Lipschitz selection existence and construction in ${f R}^2$, advancing the theory and computational methods.

Findings

Provided constructive criteria for Lipschitz selection existence.

Developed algorithms for nearly optimal Lipschitz selections.

Estimated the order of magnitude of Lipschitz seminorms.

Abstract

Let $F$ be a set-valued mapping which to each point $x$ of a metric space $({\mathcal M},\rho)$ assigns a convex closed set $F(x)\subset{\bf R}^2$. We present several constructive criteria for the existence of a Lipschitz selection of $F$, i.e., a Lipschitz mapping $f:{\mathcal M}\to{\bf R}^2$ such that $f(x)\in F(x)$ for every $x\in{\mathcal M}$. The geometric methods we develop to prove these criteria provide efficient algorithms for constructing nearly optimal Lipschitz selections and computing the order of magnitude of their Lipschitz seminorms.