A definable, p-adic analogue of Kirszbraun's Theorem on extensions of Lipschitz maps

TL;DR

This paper establishes a p-adic analogue of Kirszbraun's Theorem, showing that Lipschitz maps defined on subsets of p-adic space can be extended to the whole space in a definable manner, using p-adic geometry techniques.

Contribution

It proves that Lipschitz maps on definable subsets of p-adic space can be extended definably, extending previous real and definable p-adic results.

Findings

Existence of definable, Lipschitz retractions in p-adic space.

Extension of Lipschitz maps to the entire p-adic space while preserving definability.

Application of p-adic geometry and model theory techniques.

Abstract

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.