Kirszbraun theorem

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

${\displaystyle f:U\rightarrow H_{2}}$

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

${\displaystyle F:H_{1}\rightarrow H_{2}}$

that extends f and has the same Lipschitz constant as f.

Note that this result in particular applies to Euclidean spaces Eⁿ 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 H₁ 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 ${\displaystyle \mathbb {R} ^{n}}$ with the maximum norm and ${\displaystyle \mathbb {R} ^{m}}$ carries the Euclidean norm. ^[4] More generally, the theorem fails for ${\displaystyle \mathbb {R} ^{m}}$ equipped with any ${\displaystyle \ell _{p}}$ norm (${\displaystyle p\neq 2}$) (Schwartz 1969, p. 20). ^[2]

Explicit formulas

For an ${\displaystyle \mathbb {R} }$-valued function the extension is provided by ${\displaystyle {\tilde {f}}(x):=\inf _{u\in U}{\big (}f(u)+{\text{Lip}}(f)\cdot d(x,u){\big )},}$ where ${\displaystyle {\text{Lip}}(f)}$ is the Lipschitz constant of ${\displaystyle f}$ on U. ^[5]

In general, an extension can also be written for ${\displaystyle \mathbb {R} ^{m}}$-valued functions as ${\displaystyle {\tilde {f}}(x):=\nabla _{y}({\textrm {conv}}(g(x,y))(x,0)}$ where ${\displaystyle g(x,y):=\inf _{u\in U}\left\{\langle f(u),y\rangle +{\frac {{\text{Lip}}(f)}{2}}\|x-u\|^{2}\right\}+{\frac {{\text{Lip}}(f)}{2}}\|x\|^{2}+{\text{Lip}}(f)\|y\|^{2}}$ 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

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

External links

• Kirszbraun theorem at Encyclopedia of Mathematics.