A constructive proof of Kirszbraun's theorem
In what follows by \(\mathcal{X}\) we denote an \(n\)-dimensional Euclidean, spherical or hyperbolic space. A \(PL\)-isometry \(\varphi:P\rightarrow X\) of a polytope \(P\) in \(\mathcal{X}\) is defined as a continuous map for which there exits a locally finite triangulation of \(P\) such that, for any simplex \(T\) from this triangulation, the restriction of \(\varphi\) to \(T\) is a motion. The authors prove that any weakly contractive map \(f\) on a finite set of point \(U\subseteq \mathcal{X}\) can be extended to a \(PL\)-isometry on the entire space.
scientific article; zbMATH DE number 5941757