Volume of polyhedron

Question

Given the following polyhedron: All the $n\times n$ matrices $\boldsymbol{X}$ with elements $x_{ij}\in(0,1)$ such that $$\boldsymbol{X}\cdot\boldsymbol{1}=\boldsymbol{r}, \boldsymbol{1}^T\boldsymbol{X}=\boldsymbol{c}^T$$

For some given vectors $\boldsymbol{r}$ and $\boldsymbol{c}$.

Can I calculate the volume of this polyhedron? Can I calculate it's surface?

Answer 1

When $\boldsymbol{r}=\boldsymbol{c}=(1,1,\dots,1)$ the polytope $X$ is the Birkhoff polytope of $n\times n$ doubly-stochastic matrices. Computing its $(n-1)^2$-dimensional volume is a well-known difficult problem. An answer is given by De Loera, Liu, and Yoshida in http://arxiv.org/abs/math/0701866, but it is quite a complicated formula. For generalizing it to $\boldsymbol{r}$ and $\boldsymbol{c}$ and to higher dimension, see De Loera's survey at http://arxiv.org/pdf/1307.0124.pdf. For additional information in the case of the Birkhoff polytope, see OEIS A078524, A078525, A037302.

Answer 2

The volume can be approximated by MCMC algorithms, see https://arxiv.org/abs/1312.2873 for estimations for $n\leq 15$. Also see https://arxiv.org/abs/0705.2422 for an asymptotic formula.