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).

**Added at Edit.** Thanks to the comment of @Wlod AA, I have found an answer to my question on page 32 of the book of Benyamini and Lindenstrauss. They write that Isbell in 1964 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.

Isbell in 1964 suggested the construction of the injective envelope of a metric space, which is the smallest 1-Lipschitz AR containing a given metric space. You may have an easier time to read "Linearization..." from Bull.Acad.Polon.Sci.**16** (1968), pp. 189-193. This paper defines/constructs the metric envelope (and proves its properties), and it proves that the metric envelope of a Banach space is the metric Banach envelope. $\endgroup$9more comments