Existence Criteria for Lipschitz Selections of Set-Valued Mappings in \bf R²

2023/06/24 by Pavel Shvartsman, Shvartsman, Pavel
Computer Science · Mathematics · #46E35 #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.2306.14042

openalex publication_date 2023/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F be a set-valued mapping which to each point x of a metric space (\mathcal M,ρ) assigns a convex closed set F(x)⊂\bf R². We present several constructive criteria for the existence of a Lipschitz selection of F, i.e., a Lipschitz mapping f:\mathcal M→\bf R² such that f(x)∈ F(x) for every x∈\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.

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