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

الأصل · EN

A direct application of Zorn's Lemma gives that every Lipschitz map f:X⊂ Qₚⁿ→ Qₚℓ has an extension to a Lipschitz map f: Qₚⁿ→ Qₚℓ. This is analogous, but more easy, to Kirszbraun's Theorem about the existence of Lipschitz extensions of Lipschitz maps S⊂ Rⁿ→ Rℓ. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun's Theorem. In the present paper, we prove in the p-adic context that 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 Qₚⁿ to the topological closure of X when X is definable.