Two-sided Kirszbraun theorem

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 f : Y → ℝ^m′ that does not decrease distances more than “necessary”. Namely, ∥f(x) − f(y)∥ ≥ c√ε min(∥x − y∥, _a,b inf _∈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∥, 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 ∥f(x) − f(y)∥ between images of points x, y ∈ Y rather than in the map f 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 ∥f(x) − f(y)∥.