Two-Sided Kirszbraun Theorem

Authors Arturs Backurs, Sepideh Mahabadi, Konstantin Makarychev, Yury Makarychev

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Arturs Backurs
  • Toyota Technological Institute at Chicago (TTIC), IL, USA
Sepideh Mahabadi
  • Toyota Technological Institute at Chicago (TTIC), IL, USA
Konstantin Makarychev
  • Northwestern University, Evanston, IL, USA
Yury Makarychev
  • Toyota Technological Institute at Chicago (TTIC), IL, USA

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Arturs Backurs, Sepideh Mahabadi, Konstantin Makarychev, and Yury Makarychev. Two-Sided Kirszbraun Theorem. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In this paper, we prove a two-sided variant of the Kirszbraun theorem. Consider an arbitrary subset X of Euclidean space and its superset Y. Let f be a 1-Lipschitz map from X to ℝ^m. The Kirszbraun theorem states that the map f can be extended to a 1-Lipschitz map ̃ f from Y to ℝ^m. While the extension ̃ f does not increase distances between points, there is no guarantee that it does not decrease distances significantly. In fact, ̃ f may even map distinct points to the same point (that is, it can infinitely decrease some distances). However, we prove that there exists a (1 + ε)-Lipschitz outer extension f̃:Y → ℝ^{m'} that does not decrease distances more than "necessary". Namely, ‖f̃(x) - f̃(y)‖ ≥ c √{ε} min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) for some absolutely constant c > 0. This bound is asymptotically optimal, since no L-Lipschitz extension g can have ‖g(x) - g(y)‖ > L min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) even for a single pair of points x and y. In some applications, one is interested in the distances ‖f̃(x) - f̃(y)‖ between images of points x,y ∈ Y rather than in the map f̃ itself. The standard Kirszbraun theorem does not provide any method of computing these distances without computing the entire map ̃ f first. In contrast, our theorem provides a simple approximate formula for distances ‖f̃(x) - f̃(y)‖.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing
  • Kirszbraun theorem
  • Lipschitz map
  • Outer-extension
  • Two-sided extension


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