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? 



Volume of polyhedron
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.


