TITLE: Explicit extension of Lipschitz function (Kirszbraun theorem)
QUESTION [17 upvotes]: 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 $\sqrt{n}$.

REPLY [4 votes]: 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 $$f_*(x):=\cap_{u\in U}\overline{B}(f(u),L)$$
(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. 
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.