A metric space $X$ is called an *absolute $L$-Lipschitz retract* if for any metric space $Y$ containing $X$ there exists a Lipschitz retraction $r:Y\to X$ with Lipschitz constant $Lip(r)\le L$.


**Question.** Is each compact metric space isometric to a subset of a compact absolute 1-Lipschitz retract?


**Remark 1.** Using almost isometric embeddings into the Banach space $c\_0$, it can be shown that each compact metric space $X$ is a subset of a compact absolute $(1+\varepsilon)$-Lipschitz retract $Y$, where $\varepsilon$ is any positive real number. The space $Y$ is a suitable cube $\prod\_{n\in\omega}[-a\_n,a\_n]$ in $c\_0$ with a bit distorted metric. 


**Remark 2.** There exists also a functorial construction of an embedding of compact metric space $X$ into a compact absolute 8-Lipschitz retract $A(X)$. Given a compact metric space $X$, consider the isometric embedding $X\subset\ell\_\infty$ identifying each point $x\in X$ with the distance function $d\_X(x,\cdot)$. Next, take the closed convex hull $conv(X)$ of $X$ in $\ell\_\infty$. Finally, consider the hyperspace $A(X)$ of non-empty convex compact subsets of $conv(X)$, endowed with the Hausdorff metric. By Theorem 1.7 in the book "Geometric Nonlinear Functional Analysis" by Benyamini and Lindenstrauss, the compact metric space $A(X)$ is an absolute 8-Lipschitz retract. I do not know if the constant 8 can be replaced by a smallest constant (say 1).




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**Added at Edit.** Thanks to the comment of @Wlod AA, I have found an answer to my question on [page 32](https://books.google.com/books?id=lXZ95EKwjYUC&pg=PA32) of the book of Benyamini and Lindenstrauss. They write that [Isbell in 1964](https://eudml.org/doc/139281) suggested the construction of the injective envelope of a metric space, which is the smallest 1-Lipschitz AR containing a given metric space. For a compact metric space its injective envelope is compact, too.



Is each compact metric space a subset of a compact absolute 1-Lipschitz retract?
The classical paper on the given theme is by [Aronszajn and Panitchkpakdi](https://projecteuclid.org/euclid.pjm/1103043960). I am pretty sure that it contains the required result about embedding metric spaces into *metric* absolute retracts, i.e. in Lip$\_1$ category, and perhaps about embeddings compact metric spaces in compact metric absolute retracts. Indeed, these authors have introduced the notion of the hyper-convex metric spaces which is equivalent to metrically injective spaces (i.e. in Lip$\_1$ category, which I call simply *metric category* or *Met* for short).


This and further results are contained in later papers by John Isbell and (a bit later and independently) by wh.




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Let me provide perhaps the simplest embedding of arbitrary compact metric space $\ \mathbf X:=(X\ d)\ $ into a compact metric absolute retract.


Let $\ Y:=\mbox{Met}\_\delta\ (\mathbf X)\ $ be the set of all metric maps
$\ f:X\rightarrow[0;\delta],\ $ where $\ \delta\ $ is the diameter of $\ \mathbf X\ $ (*metric maps* means Lip$\_1).\ $ Then $\ Y\ $ in its uniform distance function is compact, it's hyper-convex i.e. it's an injective metric space (metric absolute retract), and the embedding $\ i:X\rightarrow Y\ $ is given by Kuratowski-Wojdysławski formula:


$$ \forall\_{s\ t\in X}\ \ (i(s))(t)\ :=\ d(s\ t) $$



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**Theorem** Let $\ \mathbf X:=(X\ d)\ $ be an arbitrary non-empty metric space of an arbitrary finite diameter. Let
 $\ -\infty\le a\le b\le\infty.\ $


Then space
 $\ Y\ \:=\ \mbox{Met}(\mathbf X\,\ \mathbb R\!\cap\![a;b])\ $
of all metric functions $\ f:X\rightarrow\mathbb R\cap[a;b]\ $
is hyper-convex.


**Proof**   Let $\ \emptyset\ne F\subseteq Y $
and *radia* $\ r:F\rightarrow[0;\infty)\ $ be such that


$$ \forall\_{f\ g\in F}\ r\_f+r\_g\ge |f-g| $$


Define $\ c : X\rightarrow \mathbb R\!\cap\![a;b]\ $ as follows:


$$ \forall\_{x\in X}\ \ c(x)\ :=\ \max(a\ \ sup\_{\_{x\in X}}\ (f(x)-r\_f)) $$


Then, by routine applications of the triangle inequality, the function $\ c\ $ has what it takes:


$$ c\ \in Y\cap\bigcap\_{f\in F} B(f\ r\_f) $$


where $\ B(f\ r\_f)\ $ is the ball centered in $\ f,\ $ of radius $\ r\_f.$


