KirszbraunSätze

Kirszbraun-Sätze, or Kirszbraun’s extension theorem, is a result in metric geometry about extending Lipschitz maps from a subset of a Euclidean or Hilbert space to the whole space without increasing the Lipschitz constant. It is named after Frigyes Kirszbraun, who established the original Euclidean version in 1930.

Statement for Euclidean spaces: Let A be a subset of R^m and f: A -> R^n be Lipschitz with constant L, meaning ||f(x) - f(y)|| <= L ||x - y|| for all x, y in A. Then there exists an extension F: R^m -> R^n with Lip(F) = L (i.e., ||F(x) - F(y)|| <= L ||x - y|| for all x, y in R^m) and F|_A = f. The same property holds with the target being R^n, and the extension preserves the optimal Lipschitz constant.

Generalization to Hilbert spaces: The theorem remains true when A is a subset of a real Hilbert space H1 and f: A -> H2 is Lipschitz with constant L; there exists F: H1 -> H2 with Lip(F) = L and F|_A = f.

Remarks: The constant L cannot generally be improved, and Kirszbraun’s theorem fails in general Banach spaces without inner-product structure. Extensions to certain metric spaces exist but require additional conditions.

History and applications: The result, attributed to Kirszbraun, is foundational in Lipschitz analysis with applications to interpolation, data fitting, geometric analysis, and embedding problems. See also related extension results such as the McShane extension theorem and Whitney extension theorem.