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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.

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.

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 $\ \le\ \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\ge 2\ $ we get

$$ n+2\ \le\ u(n)\ \le U(n)\ \le\ \big\lceil\frac{3\cdot n}2\big\rceil + 1 $$

We see from (i)+(ii) that finding $\ u(n)\ $ got reduced to a finite computation.

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.

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 $\ \le\ \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\ge 2\ $ we get

$$ n+2\ \le\ u(n)\ \le U(n)\ \le\ \big\lceil\frac{3\cdot n}2\big\rceil + 1 $$

We see from (i)+(ii) that finding $\ u(n)\ $ got reduced to a finite computation.

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 $\ \le\ \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\ge 2\ $ we get

$$ n+2\ \le\ u(n)\ \le U(n)\ \le\ \big\lceil\frac{3\cdot n}2\big\rceil + 1 $$

We see from (i)+(ii) that finding $\ u(n)\ $ got reduced to a finite computation.