Real Algebraic and Analytic Geometry Preprint Server

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Submission: 2009, June 5.

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.

Mathematics Subject Classification (2000): 03C64, 52A35, 32B20.

Keywords and Phrases: Kirszbraun's Theorem, Helly's Theorem, Lipschitz maps, definably complete structures.

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