Explicit extension of Lipschitz function (Kirszbraun theorem)

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.)

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.