We study various aspects of nonexpansive retracts and retractions in certain Banach and metric spaces, with special emphasis on the compact nonexpansive envelope property.

Let $(X,d)$, $(Y,\rho )$ be metric spaces and $f:X\to Y$ an injective mapping. We put ${\parallel f\parallel }_{Lip}=sup\{\rho (f\left(x\right),f\left(y\right))/d(x,y);x,y\in X,x\ne y\}$, and $dist\left(f\right)={\parallel f\parallel }_{Lip}.{\parallel f^{-1}\parallel }_{Lip}$ (the distortion of the mapping $f$). We investigate the minimum dimension $N$ such that every $n$-point metric space can be embedded into the space ${\ell }_{\infty }^N$ with a prescribed distortion $D$. We obtain that this is possible for $N\ge C{(logn)}^2{n}^{3/D}$, where $C$ is a suitable absolute constant. This improves a result of Johnson, Lindenstrauss and Schechtman [JLS87] (with a simpler proof). Related results for embeddability into ${\ell }_p^N$ are obtained by a similar method.