  The uniform probability measure on a convex polytope induces piecewise
polynomial densities on its projections. For a fixed combinatorial type of
simplicial polytopes, the moments of these measures are rational functions in
the vertex coordinates. We study projective varieties that are parametrized by
finite collections of such rational functions. Our focus lies on determining
the prime ideals of these moment varieties. Special cases include Hankel
determinantal ideals for polytopal splines on line segments, and the relations
among multisymmetric functions given by the cumulants of a simplex. In general,
our moment varieties are more complicated than in these two special cases. They
offer challenges for both numerical and symbolic computing in algebraic
geometry.
