A Definable -adic Analogue of Kirszbraun’s Theorem on Extensions of Lipschitz Maps
نویسندگان
چکیده
A direct application of Zorn’s lemma gives that every Lipschitz map f : X ⊂ Qp → Qp has an extension to a Lipschitz map f̃ : Qp → Qp . This is analogous to, but easier than, Kirszbraun’s theorem about the existence of Lipschitz extensions of Lipschitz maps S ⊂ Rn → R`. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun’s theorem. In this paper, we prove in the p-adic context that f̃ 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 Qp to the topological closure of X when X is definable.
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