Kirszbraun

Kirszbraun is a mathematical theorem named after the Polish mathematician Mieczysław Kirszbraun, who formulated it in the early 20th century. The theorem concerns the extension of Lipschitz continuous functions between subsets of Euclidean spaces.

The Kirszbraun theorem states that if a function \(f: A \to \mathbb{R}^m\), defined on a subset \(A\) of \(\mathbb{R}^n\), is Lipschitz continuous with Lipschitz constant \(L\), then there exists an extension \(F: \mathbb{R}^n \to \mathbb{R}^m\) that agrees with \(f\) on \(A\), remains Lipschitz continuous with the same Lipschitz constant \(L\), and is defined on the whole space \(\mathbb{R}^n\).

This theorem has significant implications in analysis and metric geometry, providing a foundation for extending Lipschitz functions without increasing their Lipschitz constant. It ensures that such functions can be extended optimally, preserving their regularity properties.

The proof of Kirszbraun's theorem relies on geometric and functional analysis techniques, often involving the concept of minimal Lipschitz extensions and convexity properties of metric spaces.

The theorem is fundamental in various fields, including approximation theory, geometric measure theory, and the theory of Banach spaces. Its applications are widespread, especially in problems involving function extension, interpolation, and the study of metric space embeddings.