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Monday, June 16th, 2014

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    Definable versions of theorems by Kirszbraun and Helly
    Matthias Aschenbrenner, Andreas Fischer
    (Submitted on 5 Jun 2009 (v1), last revised 15 Jul 2010 (this version, v2))

    Kirszbraun's Theorem states that every Lipschitz map S→Rn, where S⊆Rm, has an extension to a Lipschitz map Rm→Rn with the same Lipschitz constant. Its proof relies on Helly's Theorem: every family of compact subsets of Rn, having the property that each of its subfamilies consisting of at most n+1 sets share a common point, has a non-empty intersection. We prove versions of these theorems valid for definable maps and sets in arbitrary definably complete expansions of ordered fields.
    Кстати вот теорема:

    Есть непрерывное отображение единичной m-сферы в R^n. Тогда есть слой, (m-n)-объем которого не меньше объема (m-n)-сферы.
    Называется waist-inequality.

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