Polyhedra are widely used in program analysis as a means of representing (an overapproximation of) set of all possible states, where a state records the value of each variable in the program. If there are a set of invariants of the program variables, each of which can be represented as a linear inequality on the variables, then conjoining all of these invariants yields a polyhedron. If $s$ is a possible state of the program, then $s$ will be in the interior of the polyhedron (but not necessarily the reverse).
In computer security, work on quantitative information flow has applied these methods to estimate the amount of confidential information that might be leaked by a particular program. Here we build a polyhedron representing possible states of the program at a particular point in its execution, and then we want to estimate something about the number of possible states (this is related to the amount of information released). Thus, at a certain point in the analysis, they end up trying to count the number of integer points contained inside the polyhedron. This smells related to volume estimation (to me).
Here is an early paper that is representative:
That said, this might not be exactly what you are looking for. It requires methods to count the number of integer points inside the polyhedron, which isn't the same as the volume of the polyhedron. Also, I don't think they need to analyze polyhedra of dimension 1000 or higher (though I'm not sure about that).