Kirszbraun theorem

{{short description|Mathematical theorem related to real and functional analysis}}

In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if {{mvar|U}} is a subset of some Hilbert space {{mvar|H{{sub|1}}}}, and {{mvar|H{{sub|2}}}} is another Hilbert space, and

: $f: U \rightarrow H_2$

is a Lipschitz-continuous map, then there is a Lipschitz-continuous map

: $F: H_1 \rightarrow H_2$

that extends {{mvar|f}} and has the same Lipschitz constant as {{mvar|f}}.

Note that this result in particular applies to Euclidean spaces {{math|E{{sup|n}}}} and {{math|E{{sup|m}}}}, and it was in this form that Kirszbraun originally formulated and proved the theorem.{{cite journal |first=M. D. |last=Kirszbraun |title=Über die zusammenziehende und Lipschitzsche Transformationen |journal=Fundamenta Mathematicae |volume=22 |pages=77–108 |year=1934 |doi=10.4064/fm-22-1-77-108 |doi-access=free }} The version for Hilbert spaces can for example be found in (Schwartz 1969, p. 21).{{cite book |author-link=Jack Schwartz |first=J. T. |last=Schwartz |title=Nonlinear functional analysis |publisher=Gordon and Breach Science |location=New York |year=1969 }} If {{mvar|H{{sub|1}}}} 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.{{cite journal |first=D. H. |last=Fremlin |year=2011 |title=Kirszbraun's theorem |journal=Preprint |url=https://www1.essex.ac.uk/maths/people/fremlin/n11706.pdf }}

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 $\mathbb{R}^n$ with the maximum norm and $\mathbb{R}^m$ carries the Euclidean norm.{{cite book |first=H. |last=Federer |title=Geometric Measure Theory |url=https://archive.org/details/geometricmeasure00fede_0 |url-access=registration |publisher=Springer |location=Berlin |year=1969 |page=[https://archive.org/details/geometricmeasure00fede_0/page/202 202] }} More generally, the theorem fails for $\mathbb{R}^m$ equipped with any $\ell_p$ norm ( $p \neq 2$ ) (Schwartz 1969, p. 20).

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 {{mvar|U}}.{{Cite journal |last=McShane |first=E. J. |date=1934 |title=Extension of range of functions |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-40/issue-12/Extension-of-range-of-functions/bams/1183497871.full |journal=Bulletin of the American Mathematical Society |volume=40 |issue=12 |pages=837–842 |doi=10.1090/S0002-9904-1934-05978-0 |issn=0002-9904|doi-access=free }}

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.{{Cite journal |last1=Azagra |first1=Daniel |last2=Le Gruyer |first2=Erwan |last3=Mudarra |first3=Carlos |date=2021 |title=Kirszbraun's Theorem via an Explicit Formula |journal=Canadian Mathematical Bulletin |language=en |volume=64 |issue=1 |pages=142–153 |doi=10.4153/S0008439520000314 | doi-access=free |issn=0008-4395|arxiv=1810.10288 }}

History

The theorem was proved by Mojżesz David Kirszbraun, and later it was reproved by Frederick Valentine,{{cite journal |first=F. A. |last=Valentine |title=A Lipschitz Condition Preserving Extension for a Vector Function |journal=American Journal of Mathematics |volume=67 |issue=1 |year=1945 |pages=83–93 |doi=10.2307/2371917 |jstor=2371917 }} who first proved it for the Euclidean plane.{{cite journal |first=F. A. |last=Valentine |title=On the extension of a vector function so as to preserve a Lipschitz condition |journal=Bulletin of the American Mathematical Society |volume=49 |pages=100–108 |year=1943 |issue=2 |mr=0008251 |doi=10.1090/s0002-9904-1943-07859-7|doi-access=free }} Sometimes this theorem is also called Kirszbraun–Valentine theorem.

References

External links

[https://www.encyclopediaofmath.org/index.php/Kirszbraun_theorem Kirszbraun theorem] at Encyclopedia of Mathematics.

Category:Lipschitz maps

Category:Metric geometry

Category:Theorems in real analysis

Category:Theorems in functional analysis

Category:Hilbert spaces