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

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.