Kirszbraun theorem

In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if $U$ is a subset of some Hilbert space $H 1$ , and $H 2$ 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 $f$ and has the same Lipschitz constant as $f$ .

Note that this result in particular applies to Euclidean spaces $E n$ and $E m$ , and it was in this form that Kirszbraun originally formulated and proved the theorem.^[1] The version for Hilbert spaces can for example be found in (Schwartz 1969, p. 21).^[2] If $H 1$ 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.^[3]

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.^[4] More generally, the theorem fails for $\mathbb {R} ^{m}$ equipped with any $\ell _{p}$ norm ( $p\neq 2$ ) (Schwartz 1969, p. 20).^[2]

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 $U$ .^[5]

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.^[6]

History

The theorem was proved by Mojżesz David Kirszbraun, and later it was reproved by Frederick Valentine,^[7] who first proved it for the Euclidean plane.^[8] Sometimes this theorem is also called Kirszbraun–Valentine theorem.

References

^ Kirszbraun, M. D. (1934). "Über die zusammenziehende und Lipschitzsche Transformationen". Fundamenta Mathematicae. 22: 77–108. doi:10.4064/fm-22-1-77-108.
^ ^a ^b Schwartz, J. T. (1969). Nonlinear functional analysis. New York: Gordon and Breach Science.
^ Fremlin, D. H. (2011). "Kirszbraun's theorem" (PDF). Preprint.
^ Federer, H. (1969). Geometric Measure Theory. Berlin: Springer. p. 202.
^ McShane, E. J. (1934). "Extension of range of functions". Bulletin of the American Mathematical Society. 40 (12): 837–842. doi:10.1090/S0002-9904-1934-05978-0. ISSN 0002-9904.
^ Azagra, Daniel; Le Gruyer, Erwan; Mudarra, Carlos (2021). "Kirszbraun's Theorem via an Explicit Formula". Canadian Mathematical Bulletin. 64 (1): 142–153. arXiv:1810.10288. doi:10.4153/S0008439520000314. ISSN 0008-4395.
^ Valentine, F. A. (1945). "A Lipschitz Condition Preserving Extension for a Vector Function". American Journal of Mathematics. 67 (1): 83–93. doi:10.2307/2371917. JSTOR 2371917.
^ Valentine, F. A. (1943). "On the extension of a vector function so as to preserve a Lipschitz condition". Bulletin of the American Mathematical Society. 49 (2): 100–108. doi:10.1090/s0002-9904-1943-07859-7. MR 0008251.

External links

Kirszbraun theorem at Encyclopedia of Mathematics.

[1] Kirszbraun, M. D. (1934). "Über die zusammenziehende und Lipschitzsche Transformationen". Fundamenta Mathematicae. 22: 77–108. doi:10.4064/fm-22-1-77-108.

[Schwartz1969-2] Schwartz, J. T. (1969). Nonlinear functional analysis. New York: Gordon and Breach Science.

[3] Fremlin, D. H. (2011). "Kirszbraun's theorem" (PDF). Preprint.

[4] Federer, H. (1969). Geometric Measure Theory. Berlin: Springer. p. 202.

[5] McShane, E. J. (1934). "Extension of range of functions". Bulletin of the American Mathematical Society. 40 (12): 837–842. doi:10.1090/S0002-9904-1934-05978-0. ISSN 0002-9904.

[6] Azagra, Daniel; Le Gruyer, Erwan; Mudarra, Carlos (2021). "Kirszbraun's Theorem via an Explicit Formula". Canadian Mathematical Bulletin. 64 (1): 142–153. arXiv:1810.10288. doi:10.4153/S0008439520000314. ISSN 0008-4395.

[7] Valentine, F. A. (1945). "A Lipschitz Condition Preserving Extension for a Vector Function". American Journal of Mathematics. 67 (1): 83–93. doi:10.2307/2371917. JSTOR 2371917.

[8] Valentine, F. A. (1943). "On the extension of a vector function so as to preserve a Lipschitz condition". Bulletin of the American Mathematical Society. 49 (2): 100–108. doi:10.1090/s0002-9904-1943-07859-7. MR 0008251.

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