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Kirszbraun’s theorem via an explicit formula. (English) Zbl 1521.47087

The problem of extending maps from a smaller to a larger set, keeping their properties, is of utmost importance and has an impact on many application-oriented questions. Thus, M. D. Kirszbraun [Fundam. Math. 22, 77–108 (1934; Zbl 0009.03904)] proved that a Lipschitz map \(f: E\to\mathbb{R}^n\) admits an extension from an arbitrary subset \(E\subseteq\mathbb{R}^m\) to a map \(F: \mathbb{R}^m\to\mathbb{R}^n\) with the same minimal Lipschitz constant. Subsequently, this result was generalized to maps between two Hilbert spaces by F. A. Valentine [Am. J. Math. 67, 83–93 (1945; Zbl 0061.37507)]. However, these results have the flaw that they are not constructive and not transparent.
In this interesting paper, the authors give an explicit formula for the extension in the Kirszbraun-Valentine theorem and discuss related questions.

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
52A41 Convex functions and convex programs in convex geometry
54C20 Extension of maps
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References:

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