Explicit extension of Lipschitz function (Kirszbraun theorem)

Question

Kirszbraun theorem states that if $U$ is a subset of some Hilbert space $H_1$, and $H_2$ is another Hilbert space, and $f : U \to H_2$ is a Lipschitz-continuous map, then $f$ can be extended to a Lipschitz function on the whole space $H_1$ with the same Lipschitz constant.

Now let's take $H_2$ to be the Euclidean space $\mathbb{R}^n$. My question is: Is there way to explicitly construct this extension? Note that the standard proof (e.g. see Federer's geometric measure theory book or Schwartz's nonlinear functional analysis book) is an existence proof, which uses Hausdorff's maximal principle.

Some remarks:
1) For $n = 1$, the extension can be constructed explicitly, which works even if $H_1$ is only a metric space (with metric $d$): $\tilde{f}(x) = \inf_{y \in U} \{ f(y) + {\rm Lip}(f) d(x,y) \}$. See for example Mattila's book p. 100.

2) For $n > 1$, performing the above extension for each component of $f$ results in blowing up the Lipschitz constant by a factor of $\sqrt{n}$.

Answer 1

I like a recent proof by Akopyan and Tarasov:

A. V. Akopyan, A. S. Tarasov, "A constructive proof of Kirszbraun's theorem"(Russian), Mat. Zametki 84 (2008), no. 5, 781--784; translation in Math. Notes 84 (2008), no. 5-6, 725–728; MR2500644.

I could not find this paper in the open web, but there is a copy behind a paywall: https://dx.doi.org/10.1134/S000143460811014X

What they do: if $U\subset\mathbb R^n$ is a finite set and $f:U\to\mathbb R^n$ is 1-Lipschitz, then they construct a piecewise-linear piecewise-isometric (and hence 1-Lipshitz) extension of $f$ to the whole space. The construction is explicit, but some combinatorics is involved, so I'm not sure how it works for an infinite $U$. (I haven't read the paper but learned the proof from a seminar talk by one of the authors.)

Answer 2

A recent work Kirszbraun's theorem via an explicit formula by Daniel Azagra, Erwan Le Gruyer, Carlos Mudarra at https://arxiv.org/abs/1810.10288 extends the explicit formula $\tilde f$ given in the question for N=1 to any Hilbert spaces.

Answer 3

If I remember well the Kirszbraun's extension of a $L$-Lipschitz map $f:U\subset H_1\to H_2$ has the following canonical construction, analogous to the one-dimensional case you mentioned (so in a sense it is explicit).

Let $\mathcal{Co} (H_2)$ denote the metric space of all non-empty bounded closed convex sets of $H_2$ endowed with the Hausdorff distance. Let $f_*:H_1\to \mathcal{Co}(H_2)$ be defined by $$f_*(x):=\cap_{u\in U}\overline{B}(f(u),L\|x-u\|)$$

(In other words, $f_*$ takes $x\in H_1$ to the set of the admissible values at $x$ for any $L$-Lipschitz extension of $f$ to $U\cup\{x\}$). This map $f_*$ has the same Lipschitz constant of $f$, w.r.to the Hausdorff distance on $\mathcal{Co}(H_2)$.

Any non-empty bounded closed convex $C$ of a Hilbert space $H$ has a well-defined point $\kappa( C), $ the center of the closed ball of minimum radius containing $C$; this point is unique, and the corresponding map $\kappa: \mathcal{Co}(H)\to H $ is $1$-Lipschitz. (Warning - here there is an issue; it seems this is not the right selection map; see Junekey Jeon‘s comment below)

One can therefore define a canonical $L$-Lipschitz extension of $f$ as $\tilde f:=\kappa \circ f_*$. In case $n=1$, the set $f_*(x)$ is just an interval, its end-points are the inf-convolution you mentioned, and the sup-convolution, and this $\tilde f$ is their arithmetic mean.

Answer 4

See also an efficient algorithm for constructing the Kirszbraun extension, here: https://arxiv.org/abs/1905.11930