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A DEFINABLE
$p$-ADIC ANALOGUE OF KIRSZBRAUN’S THEOREM ON EXTENSIONS OF LIPSCHITZ MAPS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 17 / Issue 1 / February 2018
- Published online by Cambridge University Press:
- 20 October 2015, pp. 39-57
- Print publication:
- February 2018
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- Article
- Export citation
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A direct application of Zorn’s lemma gives that every Lipschitz map
$f:X\subset \mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$ has an extension to a Lipschitz map 
$\widetilde{f}:\mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$. This is analogous to, but easier than, Kirszbraun’s theorem about the existence of Lipschitz extensions of Lipschitz maps 
$S\subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{\ell }$. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun’s theorem. In this 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.
