I have a Linear Constraint Satisfaction Problem i.e. I have variables $ x_1, x_2,...,x_m$, with corresponding domains $D_1,D_2,...,D_m $ satisfying linear constraints $C_1, C_2,...,C_n$ with $n << m$. I am interested in calculating the Volume in $n$-dimensional space of the region formed by $n$ particular variables, say $x_1,...,x_n$ in satisfying the problem. Due to the way the problem is formulated in the first place, I know this $n$-dimensional region, call it $\mathcal{K}$, is a closed convex polytope bounded within a unit box.

In order to compute the volume of this convex polytope, I imagine one could solve for the corners of the full problem in $m$-dimensional space and compute the volume of the convex hull of these corner points in the much smaller $n$-dimensional space, but I feel that this cannot be the most efficient method. Even if this is the best method, can someone suggest some papers for this approach?

I have considered using (Multiphase?) MC methods, but if I generate points in $\mathbb{R}^n$, in order for me to check whether it lies within my region, I have to check if there exists atleast one feasible point in $m$-D with this instantiation for $x_1,...,x_n$. So the Separation Oracle call is not cheap.

Is anyone aware of how such problems are handled or do you have any ideas for how one may approach this? Any suggestions/directions will be very helpful. Thanks!