Kirszbrauni

Kirszbrauni, better known in mathematics as Kirszbraun's extension theorem, is a result about extending Lipschitz maps. It states that if a function is defined on a subset of Euclidean space and has a given Lipschitz constant, then it can be extended to the whole space without increasing that constant. In practice, this means distances between points are not distorted by more than the original Lipschitz bound when extending the function.

Formal statement: Let A be a subset of R^n and f: A -> R^m be L-Lipschitz, meaning ||f(x) - f(y)|| ≤ L ||x - y|| for all x,y in A. Then there exists F: R^n -> R^m such that F|_A = f and the Lipschitz constant of F is also L (Lip(F) = Lip(f)). The extension preserves the original bound exactly, not merely approximately.

History and generalizations: The theorem was established by J. Kirszbraun in 1934 for Euclidean spaces. It has since been extended to maps between Hilbert spaces and, in various forms, to broader metric-space settings. The result is commonly referred to as Kirszbraun's theorem or the Kirszbraun–Toeplitz extension principle, reflecting its central role in Lipschitz extension theory.

Applications and related results: Kirszbraun's theorem is used in analysis, geometry, and metric embedding problems, notably to extend partial Lipschitz data without increasing the bound and to study distortions in embeddings of finite metric spaces. Related concepts include the McShane and Whitney extension theorems for real-valued or smooth functions, and general Lipschitz extension problems in metric spaces.

References: J. Kirszbraun, Über die Abbildung durch eine Lipschitz-Funktion, Fundamenta Mathematicae (1934). For extensions and generalizations, see surveys on Lipschitz extension theorems in Hilbert and metric spaces.