Talk:Kirszbraun theorem

Is this really true? I think that the assumption that $E^{n}$ be metrized with the $L^{\infty }$ norm has been dropped! See the article from 1956 by Aronszajn and Panitchpakdi, "Extension of uniformly continuous mappings and hyperconvex spaces", Pacific Journal of Mathematics (I can't remember the volume number).

No it is ok, the metric is Euclidean Tosha 20:41, 8 Nov 2004 (UTC)

And where can I find the proof? mbork 09:00, 2004 Nov 15 (UTC)

Well, the theorem is true for maps from (a subset of) any metric space to $l^{\infty }(\Gamma )$ (in particular, to $E^{n}$ with the supremum norm) - see below. On the other hand, the main article comments there is a counter-example using $l^{\infty }$ norm in the domain $R^{n}$. Therefore I think you perhaps just confused domain and range spaces. 147.231.88.128 (talk) 22:56, 20 November 2010 (UTC)

Now the References on the main page should be enough to find a proof.

There is a good extension of this result to metric spaces: If S is a subset of a metric space (X,d), and if f:S→R is K-Lipschitz, then f:X→R defined by

f(x):=sup{f(y)-Kd(x,y)|y∈S}

equals f on S and is K-Lipschitz. It is easy to check that f works. The result is from E. J. McShane, Extensions of Range of Functions, Bull. Am. Math. Soc., 40 1934, 837-842. It is interesting that these two papers were published in the same year. I don't know the history. -Craig Calcaterra

Yes, this version is easy. And yes, McShane's name is used by a group of people to refer to it. A preprint (http://www.maths.manchester.ac.uk/raag/preprints/0286.ps.gz) attributes it to McShane and Whitney (independently 1934) and notes it can be immediately (indeed: work coordinate-wise) generalized for R^m, with Lipschitz constant multiplied by sqrt(n). The version with "the same Lipschitz constant" (this one is not easy!) is the Kirszbraun's one 147.231.88.128 (talk) 22:56, 20 November 2010 (UTC)

The theorem is true for maps from (a subset of) any metric space to $l^{\infty }(\Gamma )$ (in particular, to $E^{n}$ with the supremum norm). And it is actually an easy (though, important!) extension of the easy (McShane's) version mentioned above: extend coordinate-wise and and check (via the definition of the supremum norm), that we get the same Lipschitz constant. (Note we have supremum norm in the range, contrary to the counter-example mentioned on the main page, which has the supremum norm in the domain.) 147.231.88.128 (talk) 22:56, 20 November 2010 (UTC)