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In mathematics , the Kirszbraun theorem (also: Kirszbraun's sentence or Kirszbraun-Valentine's sentence ) is a doctrine about the continuability of Lipschitz continuous mappings; it is named after the Polish mathematician Mojżesz Dawid Kirszbraun .

sentence

Be

${\displaystyle f\colon U\to \mathbb {R} ^{m}}$

a Lipschitz continuous map with Lipschitz constant defined on a subset , then there is a Lipschitz continuous map ${\displaystyle U\subset \mathbb {R} ^{n}}$ ${\displaystyle Lip(f)}$

${\displaystyle F\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}}$

with the same Lipschitz constant

${\displaystyle Lip(F)=Lip(f)}$

and with

${\displaystyle F\mid _{U}=f.}$

example

For can be explicitly defined by ${\displaystyle m=1}$ ${\displaystyle F\colon \mathbb {R} ^{n}\to \mathbb {R} }$

${\displaystyle F(x):=\inf _{u\in U}{\big (}f(u)+{\text{Lip}}(f)\cdot d(x,u){\big )}}$

for everyone . ${\displaystyle x\in \mathbb {R} ^{n}}$

The same formula works for subsets of arbitrary metric spaces and is known in this context as McShane's Lemma . ${\displaystyle U\subset X}$ ${\displaystyle X}$

For we know no such closed formula. ${\displaystyle m>1}$

Generalizations

Kirszbraun's theorem also applies to Hilbert spaces , but not to any Banach spaces .

Let Hilbert spaces be a Lipschitz continuous mapping defined on a subset , then there is a Lipschitz continuous mapping with the same Lipschitz constant and with ${\displaystyle H_{1},H_{2}}$ ${\displaystyle f\colon U\to H_{2}}$ ${\displaystyle U\subset H_{1}}$ ${\displaystyle F\colon H_{1}\to H_{2}}$ ${\displaystyle F\mid _{U}=f.}$

literature

• M. Kirszbraun: About the contracting and Lipschitzian transformations . Find. Math. 22 (1935), 77-108. online (pdf)
• F. Valentine: A Lipschitz condition preserving extension for a vector function. Amer. J. Math. 67: 83-93 (1945). online (pdf)

Web links

• Fremlin: Kirszbraun's Theorem
• Kirszbraun Theorem (Encyclopedia of Mathematics)