pr.probability - Integrating a simple exponential over the space of matrices that define a metric

I want to interpret an $ntimes n$ matrix $D$ as a set of pairwise distances, and assume that $D$ obeys metric properties. Namely, $D_{ii} = 0$, $D_{ij} geq 0$, $D_{ij} = D_{ji}$ and $D_{ij} leq D_{ik} + D_{kj}$ for all $1 leq i,j,k leq n$. For convenience, let $bigtriangleup_n$ denote the set of such matrices.

Now, I need to integrate some "simple" functions over this set. The simplest would be an exponential. Namely, I want to compute something like $int_{bigtriangleup_n} expleft[-lambda sum_{i=1}^n sum_{j=i+1}^n D_{ij}right] d D$.

I've been able to work this out for the simplest nontrivial case: namely $n=3$. But for higher $n$, my brute force way of calculation gets really ugly. The approach I've been taking is to basically first integrate over $D_{11}, D_{12}, D_{13}, ..., D_{1n}$, all of which have no constraints... then integrate over $D_{23}$ which is just a definite integral from $|D_{12} - D_{23}|$ to $D_{12}+D_{23}$ of $exp[-lambda D_{13}]$ and then, in the general case, integrating over $D_ij$ becomes the definite integral from $max_{k neq i,j} |D_{ik}-D_{jk}|$ to $min_{kneq i,j} D_{ik}+D_{jk}$, but this is the point at which I get stuck, because these things becomes nasty quite quickly (even for just $n=4$).

At the end of the day, I've love to be able to integrate more complex functions, like a chi-square type function rather than an exponential type function, but the exponential is the most trivial case that is interesting...

To be precise, I'm looking for a closed form evaluation of the integral above, preferably with some derivation that will help me work out more complex examples.

• Next pr.probability - sum of order statistics
• Previous dg.differential geometry - How do I make the conceptual transition from multivariable calculus to differential forms?