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Preprint Number 188

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188. Matthias Aschenbrenner, Andreas Fischer
Definable versions of theorems by Kirszbraun and Helly
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Submission date: 5 June 2009.

Abstract:

Kirszbraun's Theorem, Helly's Theorem, Lipschitz maps, definably complete structures Abstract Text: 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.

Mathematics Subject Classification: 03C64, 52A35, 32B20

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