In 1972, I have developed a general theory of isometric embeddings of finite metric spaces into finite-dimensional Banach spaces, in particular, the injective case, i.e. into metrically injective spaces $\mathbb R^n$ with metrics given by $\max$. An announcement has appeared a couple of years later in AMS Notices. One of the theorems fits the present thread: For every natural $k\ n$, there is an EXPLICIT natural constant $\mu(k\ n)$ such that the following two statements are equivalent: > (i) every $k$-element metric space can be isometrically embedded into $\mathbb R^n$ with the distance given by $\max$; > (ii) every $k$-element metric space with integer distances and of diameter ${\leq} \mu(k\ n)$ can be isometrically embedded into subspace $\{0\ \ldots\ \mu(k, n)\}^n$ of $\mathbb R^n$ with the distance given by $\max$. For each natural $\ n,\ $ there exists a maximal $\ u(n)=k\ $ as above; and also $\ U(n)\ $ similar to $\ u(n)\ $ but for ALL $n$-dimensional Banach spaces TOGETHER (for each fixed dimension $\ n).\ $ Then in the non-trivial case of $\ n \geq 2\ $ we get $$ n+2 \leq u(n) \leq U(n) \leq \left\lceil\frac{3\cdot n}2\right\rceil + 1 $$ We see from (i)+(ii) that finding $u(n)$ has been reduced to a finite computation.