Kirszbraun theorem
In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if Template:Mvar is a subset of some Hilbert space Template:Mvar, and Template:Mvar is another Hilbert space, and
- <math> f: U \rightarrow H_2</math>
is a Lipschitz-continuous map, then there is a Lipschitz-continuous map
- <math>F: H_1 \rightarrow H_2</math>
that extends Template:Mvar and has the same Lipschitz constant as Template:Mvar.
Note that this result in particular applies to Euclidean spaces Template:Math and Template:Math, and it was in this form that Kirszbraun originally formulated and proved the theorem.<ref>Template:Cite journal</ref> The version for Hilbert spaces can for example be found in (Schwartz 1969, p. 21).<ref name="Schwartz1969">Template:Cite book</ref> If Template:Mvar is a separable space (in particular, if it is a Euclidean space) the result is true in Zermelo–Fraenkel set theory; for the fully general case, it appears to need some form of the axiom of choice; the Boolean prime ideal theorem is known to be sufficient.<ref>Template:Cite journal</ref>
The proof of the theorem uses geometric features of Hilbert spaces; the corresponding statement for Banach spaces is not true in general, not even for finite-dimensional Banach spaces. It is for instance possible to construct counterexamples where the domain is a subset of <math>\mathbb{R}^n</math> with the maximum norm and <math>\mathbb{R}^m</math> carries the Euclidean norm.<ref>Template:Cite book</ref> More generally, the theorem fails for <math> \mathbb{R}^m </math> equipped with any <math> \ell_p</math> norm (<math> p \neq 2</math>) (Schwartz 1969, p. 20).<ref name="Schwartz1969" />
Explicit formulasEdit
For an <math>\mathbb{R}</math>-valued function the extension is provided by <math>\tilde f(x):=\inf_{u\in U}\big(f(u)+\text{Lip}(f)\cdot d(x,u)\big),</math> where <math>\text{Lip}(f)</math> is the Lipschitz constant of <math>f</math> on Template:Mvar.<ref>Template:Cite journal</ref>
In general, an extension can also be written for <math>\mathbb{R}^{m}</math>-valued functions as <math>\tilde f(x):= \nabla_{y}(\textrm{conv}(g(x,y))(x,0)</math> where <math>g(x,y):=\inf_{u\in U}\left\{\langle f(u),y \rangle +\frac{\text{Lip}(f)}{2}\|x-u\|^{2}\right\}+\frac{\text{Lip}(f)}{2} \|x\|^{2}+\text{Lip}(f)\|y\|^{2}</math> and conv(g) is the lower convex envelope of g.<ref>Template:Cite journal</ref>
HistoryEdit
The theorem was proved by Mojżesz David Kirszbraun, and later it was reproved by Frederick Valentine,<ref>Template:Cite journal</ref> who first proved it for the Euclidean plane.<ref>Template:Cite journal</ref> Sometimes this theorem is also called Kirszbraun–Valentine theorem.