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Preprint Number 829

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829. Raf Cluckers, Florent Martin
A definable, p-adic analogue of Kirszbraun's Theorem on extensions of Lipschitz maps

Submission date: 10 Februay 2015


A direct application of Zorn's Lemma gives that every Lipschitz map f:X\subset Q_p^n --> Q_p^\ell has an extension to a Lipschitz map \widetilde f: Q_p^n --> Q_p^\ell. This is analogous, but more easy, to Kirszbraun's Theorem about the existence of Lipschitz extensions of Lipschitz maps S\subset R^n --> 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 Q_p^n to the topological closure of X when X is definable.

Mathematics Subject Classification: 03C98, 12J25 (Primary), 03C60, 32Bxx, 11S80 (Secondary)

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Full text arXiv 1502.03036: pdf, ps.

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