Definable versions of theorems by Kirszbraun and Helly

TL;DR

This paper extends classical theorems by Kirszbraun and Helly to the setting of definable maps and sets within arbitrary definably complete expansions of ordered fields, broadening their applicability.

Contribution

It introduces definable versions of Kirszbraun's and Helly's theorems applicable in more general mathematical structures beyond Euclidean spaces.

Findings

Established definable Kirszbraun's theorem for Lipschitz maps.

Proved definable Helly's theorem for families of sets.

Extended classical theorems to arbitrary definably complete expansions.

Abstract

Kirszbraun's Theorem states that every Lipschitz map $S\to\mathbb R^n$, where $S\subseteq \mathbb R^m$, has an extension to a Lipschitz map $\mathbb R^m \to \mathbb R^n$ with the same Lipschitz constant. Its proof relies on Helly's Theorem: every family of compact subsets of $\mathbb R^n$, having the property that each of its subfamilies consisting of at most $n+1$ sets share a common point, has a non-empty intersection. We prove versions of these theorems valid for definable maps and sets in arbitrary definably complete expansions of ordered fields.