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

Let $F$ be a set-valued mapping which to each point $x$ of a metric space $(\mathcal{M},\rho )$ assigns a convex closed set $F(x)\subset {\mathbf{R}}^2$. 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.