Finiteness Principles for Smooth Selection

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.

106 pages