Motivation for volume estimation
Estimating the volume of a convex polytope and the closely related task of sampling from it have applications in private data release. 

Roughly, the problem you want to solve is: given a collection of numeric valued queries on a database, come up with answers to those questions that are as close as possible to the real answers, while satisfying differential privacy. In some range of parameters, the optimal algorithm for solving this problem has a geometric description, and implementing it involves sampling from a convex polytope. See here: http://arxiv.org/pdf/0907.3754v3.pdf