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I want to interpret an $n\times n$ 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 $\int_{\bigtriangleup_n} \exp\left[-\lambda \sum_{i=1}^n \sum_{j=i+1}^n D_{ij}\right] d D$.

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 $\min_{k\neq i,j} D_{ik}+D_{jk}$, 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.

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# Integrating a simple exponential over the space of matrices that define a metric

I want to interpret an $n\times n$ 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 $\int_{\bigtriangleup_n} \exp\left[-\lambda \sum_{i=1}^n \sum_{j=i+1}^n D_{ij}\right] d D$.

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 $\min_{k\neq i,j} D_{ik}+D_{jk}$, 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...