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Continuity of extensions of Lipschitz maps

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Abstract

We establish the sharp rate of continuity of extensions of ℝm-valued 1-Lipschitz maps from a subset A of ℝn to a 1-Lipschitz maps on ℝn. We consider several cases when there exists a 1-Lipschitz extension with preserved uniform distance to a given 1-Lipschitz map. We prove that if m > 1, then a given map is 1-Lipschitz and affine if and only if such a distance preserving extension exists for any 1-Lipschitz map defined on any subset of ℝn. This shows a striking difference from the case m = 1, where any 1-Lipschitz function has such a property. Another example where we prove it is possible to find an extension with the same Lipschitz constant and the same uniform distance to another Lipschitz map v is when the difference between the two maps takes values in a fixed one-dimensional subspace of ℝm and the set A is geodesically convex with respect to a Riemannian pseudo-metric associated with v.

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Acknowledgements

The author wishes to thank Bo’az Klartag, Eva Kopecká and Vojtĕch Kaluža for useful discussions, Simeon Reich for several comments and references brought to the author’s attention and the anonymous referee for comments that allowed for an improvement of the manuscript. The financial support of St John’s College of the University of Oxford, Clarendon Fund and EPSRC is gratefully acknowledged. Part of this research was completed in Fall 2017 while the author was a member of the Geometric Functional Analysis and Application program at MSRI, supported by the National Science Foundation under Grant No. 1440140.

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  1. Mathematical Institute and St John’s College, University of Oxford, Oxford, OX2 6GG, UK

    Krzysztof J. Ciosmak

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Correspondence to Krzysztof J. Ciosmak.

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Ciosmak, K.J. Continuity of extensions of Lipschitz maps. Isr. J. Math. 245, 567–588 (2021). https://doi.org/10.1007/s11856-021-2215-0

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