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Let $F$ be a set-valued mapping which to each point $x$ of a metric space$({\mathcal M},\rho)$ assigns a convex closed set $F(x)\subset{\bf R}^2$. Wepresent several constructive criteria for the existence of a Lipschitzselection of $F$, i.e., a Lipschitz mapping $f:{\mathcal M}\to{\bf R}^2$ suchthat $f(x)\in F(x)$ for every $x\in{\mathcal M}$. The geometric methods wedevelop to prove these criteria provide efficient algorithms for constructingnearly optimal Lipschitz selections and computing the order of magnitude oftheir Lipschitz seminorms.

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