  We show that the multivariate generating function of appropriately normalized
moments of a measure with homogeneous polynomial density supported on a compact
polytope P in R^d is a rational function. Its denominator is the product of
linear forms dual to the vertices of P raised to the power equal to the degree
of the density function. Using this, we solve the inverse moment problem for
the set of, not necessarily convex, polytopes having a given set S of vertices.
Under a weak non-degeneracy assumption we also show that the uniform measure
supported on any such polytope is a linear combination of uniform measures
supported on simplices with vertices in S.
