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

f : U → H₂

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

F: H₁ → H₂

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 Rⁿ with the maximum norm and 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]

For an 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 $\text{Lip}(f)$ is f's Lipschitz constant on U.

History

The theorem was proved by Mojżesz David Kirszbraun, and later it was reproved by Frederick Valentine,^[5] who first proved it for the Euclidean plane.^[6] Sometimes this theorem is also called Kirszbraun–Valentine theorem.

References

1. Kirszbraun, M. D. (1934). "Über die zusammenziehende und Lipschitzsche Transformationen". Fund. Math. 22: 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. 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.
6. 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: 100–108. doi:10.1090/s0002-9904-1943-07859-7. MR 0008251.

External links

• Kirszbraun theorem at Encyclopedia of Mathematics.