We have an $(n-1)^2$-dimensional convex polytope, specified by a system of $(n^2-1)$ inequalities of the form $Ax \leq b$. I need to calculate the "surface area" of an $((n-1)^2-k)$ dimensional feature (a $k$-$face$?) of this polytope, which is specified by turning $k$ of those $(n^2-1)$ inequalities into equalities.
About the polytope: It's a linear transformation of the $(n-1)^2$-dimensional Birkhoff polytope, representing the set of all $n \times n$ doubly stochastic matrices. The constraint matrix $A$ is just the same linear transformation of the doubly stochastic constraints (in $(n-1)^2$-dimensions). Each of the entries of A is either $0$ or $1$ or $-1$. The RHS vector $b$ consists entirely of $0$'s and $1$'s too.
Can anyone help? I was looking for a formula in terms of maybe $A$,$k$ etc., if possible.
Thank you so much in advance for your help.