A typical two-stage process governs the selection of meal items in an Italian menu. The process admits a direct formalization in terms of choice spaces. Path-independent choice spaces are cryptomorphic to convex geometries (or antimatroids, introduced by Robert E. Jamison). There results a construction of a new convex geometry for a convex geometry (the base) given together with a family of convex geometries (the fibers) indexed by the elements of the base; we call the outcome a resolution.
We experimentally study the problem of estimating the volume of convex polytopes, focusing on H- and V-polytopes, as well as zonotopes. Although a lot of effort is devoted to practical algorithms for H-polytopes there is no such method for the latter two representations. We propose a new, practical method for all representations, which is faster on H-polytopes; it relies on Hit-and-Run (HnR) sampling, and combines a new simulated annealing method with the Multiphase Monte Carlo (MMC) approach.