Explicit computations of Fourier transforms of polyhedral cones

The Fourier transforms of polyhedral cones can be used, via Brion’s theorem, to compute various geometric quantities of polytopes, such as volumes, moments, and lattice-point counts. We present a novel method of computing these conic Fourier transforms by polynomial interpolation. Given the fact that computing volumes of polytopes is #P-hard (Dyer–Frieze [DF88]), we cannot hope for fast algorithms in the general case. However, with extra assumptions on the combinatorics of the cone, we demonstrate it is possible to compute its Fourier transform efficiently.