The Whitney extension problem and Lipschitz selections of set-valued mappings in jet-spaces

2006/01/29 by Pavel Shvartsman, Shvartsman, Pavel
Computer Science · Mathematics · #46E35 #52A35 #54C60 #54C65 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis #math.FA #msc:46E35 #msc:52A35 #msc:54C60 #msc:54C65

paper · pdf · doi:10.48550/arxiv.math/0601711

arxiv created 2006/01/29 · openalex publication_date 2006/01/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a variant of the Whitney extension problem for the space C^k,ω(Rⁿ). We identify this space with a space of Lipschitz mappings from Rⁿ into the space P_k × Rⁿ of polynomial fields on Rⁿ equipped with a certain metric. This identification allows us to reformulate the Whitney problem for C^k,ω(Rⁿ) as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of P_k × Rⁿ. We prove a Helly-type criterion for the existence of Lipschitz selections for such set-valued mappings defined on finite sets. With the help of this criterion, we improve estimates for finiteness numbers in finiteness theorems for C^k,ω(Rⁿ) due to C. Fefferman.

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