Finiteness Principles for Smooth Selection

TL;DR

This paper establishes finiteness principles for smooth and Lipschitz selection problems in Euclidean spaces, proving a longstanding conjecture and advancing understanding of constrained interpolation with potential applications in nonnegative function interpolation.

Contribution

It proves finiteness principles for $C^{m}$ and $C^{m-1,1}$-selection, confirming a conjecture on Lipschitz selections for $ eal^n$ domains, advancing the theory of constrained interpolation.

Findings

Proved finiteness principles for smooth selection problems.

Confirmed a conjecture on Lipschitz selections from 1994.

Enhanced understanding of constrained interpolation problems.

Abstract

In this paper we prove finiteness principles for $C^{m}\left( \mathbb{R}^{n}, \mathbb{R}^{D}\right) $-selection, and for $C^{m-1,1}\left( \mathbb{R}^{n}, \mathbb{R}^{D}\right) $-selection, in particular providing a proof for a conjecture of Brudyni-Shvartsman (1994) on Lipschitz selections for the case when the domain is $X = \mathbb{R}^n$. Our results raise the hope that one can start to understand constrained interpolation problems in which e.g. the interpolating function $F$ is required to be nonnegative everywhere.