On moments of a polytope

TL;DR

This paper demonstrates that the generating function of normalized moments for measures with polynomial densities on polytopes is rational, enabling solutions to inverse moment problems and decompositions into simplices.

Contribution

It introduces a rational function representation of moments for polynomial densities on polytopes and solves the inverse moment problem for polytopes with specified vertices.

Findings

The generating function is a rational function with a specific denominator structure.

The inverse moment problem can be solved for polytopes with a given set of vertices.

Uniform measures on such polytopes can be expressed as combinations of measures on simplices.

Abstract

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.