A metric on $n$ points $N$ can be represented as a vector $x \in \mathbb{R}_+^{n \choose 2}$.

For each pair of distinct $i, j \in N$, we have $d(i,j) = d(j,i) = x_{i,j}$. The set of all metrics is the set of points which lie inside the cone defined by the (triangle) inequalities:
$$x_{i,j} - x_{i,k} - x_{j,k} \leq 0$$

for each distinct triple $\{i,j,k\} \subset N$.
If we bound the cone by the inequalities $x_{i,j} + x_{i,k} + x_{j,k} \leq 2$, then we have the *metric polytope*.

Question: Are there any (lower or upper) bounds known on the volume of the metric polytope, for general n?

In the end, I am interested in estimating the size of the smallest $\epsilon$-net for the set of bounded metrics on $n$ points, which I asked about here. I would also be interested in estimates for polytopes that result in other ways of bounding the metric cone -- for example, by including the inequalities $x_{i,j} \leq 1$ for all $i,j$.