# Definable versions of theorems by Kirszbraun and Helly - Mathematics > Logic

Definable versions of theorems by Kirszbraun and Helly - Mathematics > Logic - Download this document for free, or read online. Document in PDF available to download.

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-sTheorem: every family of compact subsets of $\mathbb R^n$, having the propertythat each of its subfamilies consisting of at most $n+1$ sets share a commonpoint, has a non-empty intersection. We prove versions of these theorems validfor definable maps and sets in arbitrary definably complete expansions ofordered fields.

Author: ** Matthias Aschenbrenner, Andreas Fischer**

Source: https://arxiv.org/