# Volume of polyhedron

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?

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.
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.