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Definable versions of theorems by Kirszbraun and Helly

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.