On the Core of a Low Dimensional Set-Valued Mapping

2021/02/15 by Shvartsman, Pavel
#46E35 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2102.07609

Abstract

Let \mathfrak M=(\mathcal M,ρ) be a metric space and let X be a Banach space. Let F be a set-valued mapping from \mathcal M into the family \mathcal K_m(X) of all compact convex subsets of X of dimension at most m. The main result in our recent joint paper with Charles Fefferman (which is referred to as a "Finiteness Principle for Lipschitz selections") provides efficient conditions for the existence of a Lipschitz selection of F, i.e., a Lipschitz mapping f:\mathcal M→ X such that f(x)∈ F(x) for every x∈\mathcal M. We give new alternative proofs of this result in two special cases. When m=2 we prove it for X=\bf R², and when m=1 we prove it for all choices of X. Both of these proofs make use of a simple reiteration formula for the "core" of a set-valued mapping F, i.e., for a mapping G:\mathcal M→\mathcal K_m(X) which is Lipschitz with respect to the Hausdorff distance, and such that G(x)⊂ F(x) for all x∈\mathcal M.

Related

• The Core of a 2-Dimensional Set-Valued Mapping. Existence Criteria and Efficient Algorithms for Lipschitz Selections of Low Dimensional Set-Valued Mappings
• Sharp finiteness principles for Lipschitz selections: long version
• Sharp finiteness principles for Lipschitz selections
• Existence Criteria for Lipschitz Selections of Set-Valued Mappings in \bf R²
• A Pointwise Lipschitz Selection Theorem