Two-Sided Kirszbraun Theorem

Abstract

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. The Kirszbraun theorem states that the map f can be extended to a 1-Lipschitz map f̃ from Y to R. While the extension f̃ does not increase distances between points, there is no guarantee that it does not decrease distances significantly. In fact, f̃ 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 ′ that does not decrease distances more than “necessary”. Namely, ∥f̃(x) − f̃(y)∥ ≥ c √ εmin(∥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)∥ > Lmin(∥x− y∥, infa,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 f̃ first. In contrast, our theorem provides a simple approximate formula for distances ∥f̃(x) − f̃(y)∥. 2012 ACM Subject Classification Theory of computation → Computational geometry; Mathematics of computing