In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if U is a subset of some Hilbert space H1, and H2 is another Hilbert space, and
Explicit formulas
For an \mathbb{R}-valued function the extension is provided by \tilde f(x):=\inf_{u\in U}\big(f(u)+\text{Lip}(f)\cdot d(x,u)\big), where \text{Lip}(f) is the Lipschitz constant of f on . In general, an extension can also be written for \mathbb{R}^{m}-valued functions as \tilde f(x):= \nabla_{y}(\textrm{conv}(g(x,y))(x,0) where 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} and conv(g) is the lower convex envelope of g. ==History==
History
The theorem was proved by Mojżesz David Kirszbraun, and later it was reproved by Frederick Valentine, who first proved it for the Euclidean plane. Sometimes this theorem is also called Kirszbraun–Valentine theorem. ==References==