%0 Journal Article
%T A definable, p-adic analogue of Kirszbraun's Theorem on extensions of Lipschitz maps
%A Raf Cluckers
%A Florent Martin
%J Mathematics
%D 2015
%I arXiv
%R 10.1017/S1474748015000390
%X A direct application of Zorn's Lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_p^n\to \mathbb{Q}_p^\ell$ has an extension to a Lipschitz map $\widetilde f: \mathbb{Q}_p^n\to \mathbb{Q}_p^\ell$. This is analogous, but more easy, to Kirszbraun's Theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset \mathbb{R}^n\to \mathbb{R}^\ell$. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun's Theorem. In the present paper, we prove in the $p$-adic context that $\widetilde f$ can be taken definable when $f$ is definable, where definable means semi-algebraic or subanalytic (or, some intermediary notion). We proceed by proving the existence of definable, Lipschitz retractions of $\mathbb{Q}_p^n$ to the topological closure of $X$ when $X$ is definable.
%U http://arxiv.org/abs/1502.03036v1