I would like to make the claim that as the number of inequalities increase, the volume of the subspace of feasible values decreases at least by some ratio K in expectation. I am guessing the ratio would be a function of the distance of the hyperplanes to the "center" of the subspace.
I looked into maximally inscribed hyperspheres and tried to find some bounds on their volumes and the polytope's. I think, if such a bound exists, it can be used to establish a lower bound.