Finitness 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 $X={\mathbb{R}}^n$.

This is the first part of the paper titled "Finiteness Principles for Smooth Selection" arXiv:1511.04804 . A new corollary (Corollary 4 in Section III.1) has been added. The original paper is split into two parts for publication convenience