Kirszbraun theorem

2020 Mathematics Subject Classification: Primary: 54E40 [MSN][ZBL]

A theorem in real analysis, proved first by Kirszbraun in [Ki], which states that, if $E\subset \mathbb R^n$, then any Lipschitz function $f: E \to \mathbb R^m$ can be extended to the whole $\mathbb R^n$ keeping the Lipschitz constant of the original function. In the case $m=1$ the theorem is rather straightforward, since one such extension is given by \[ \tilde{f} (x) := \inf_{y\in E}\, (f (y) + {\rm Lip (f)} |x-y|)\, . \] In fact when the target space is $\mathbb R$, the formula above can be easily generalized to a subset $E$ of any metric space $(X,d)$: it suffices to replace $|x-y|$ with $d (x,y)$. The general case $m>1$ is instead rather complicated. For an elegant and concise proof see 2.10.43 of [Fe]. Note that a Lipschitz extension with a non-optimal constant can be easily achieved using the formula above for each component of the vector function.

The theorem remains valid if both $\mathbb R^n$ and $\mathbb R^m$ are replaced by general Hilbert spaces $H_1$ and $H_2$, see [Va]. When $H_1$ is not separable such extension requires some form of the Axiom of choice. With the exception of the trivial case when the target is $\mathbb R$, generalizations of Kirszbraun's theorem are rather delicate: it is for instance known that it does not hold if any of the two spaces $H_1$ and $H_2$ are replaced by Banach spaces. However it holds between Riemannian manifolds endowed with the geodesic distances under very special assumptions, for instance if both spaces are spheres of the same dimension, or if they have both constant curvature $-1$ (see [LS] and references therein). For a generalization to Alexandrov spaces under some suitable assumptions see the work [LS].

References

[Fe]	H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. MR0257325 Zbl 0874.49001
[Ki]	M.D. Kirszbraun, "Ueber die zusammenziehenden und Lipschitzsche Transformationen", Fund. Math. 22 (1935), 77-108.
[LS]	U. Lang, V. Schroeder, "Kirszbraun's theorem and metric spaces of bounded curvature", GAFA 7 (1997) 535-560.
[Va]	A. Valentine, "Contractions in non-Euclidean spaces", Bull. Amer. Math. Soc. 50 (1944) 710-713.