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

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 $F(x)\subset {\mathbf{R}}^2$. We present several constructive criteria for the existence of a Lipschitz selection of $F$, i.e., a Lipschitz mapping $f:\mathcal{M}\to {\mathbf{R}}^2$ 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.

arXiv comment80 pages, 20 figures