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? 
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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](https://cstheory.stackexchange.com/questions/4403/counting-metrics). 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 $i,j$.



Estimating the Volume of the Metric Polytope
Some googling reveals:


1. What you call a *metric polytope* is also called a *semimetric polytope* (see for example the standard reference: [Geometry of cuts and metrics](http://books.google.com/books?id=Lx9dYMW70J0C&lpg=PR1&ots=SAFSPMtHru&dq=deza%2520geometry&pg=PR1#v=onepage&q&f=false) )
2. In the same book, [see here](http://books.google.com/books?id=Lx9dYMW70J0C&lpg=PR1&ots=SAFSPMtHru&dq=deza%2520geometry&pg=PA427#v=onepage&q=volume%2520semimetric&f=false), the authors say that the volume of the *rooted* semimetric polytope in dimension $n+1$ is: $$\frac{n!}{(2n)!}2^n$$


**EDIT:** Further web searching (not googling) showed up the paper: [On Skeletons, Diameters and Volumes of Metric Polyhedra](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.213). In this paper, the authors give explicit volumes for $n=3$ to $n=6$, and given those volumes it seems that the volume goes very rapidly to zero (e.g., $vol\_4=2/45$, $vol\_5=4/1701$, $vol\_6=71936/1477701225$). 


At this point, my current knowledge of the volume ends!


