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arXiv · 1210.3193

On moments of a polytope

Nick Gravin·Dmitrii V. Pasechnik·Boris Shapiro·Michael Shapiro
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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.

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Nick Gravin, Dmitrii V. Pasechnik, Boris Shapiro, Michael Shapiro. 2017-11-29. On moments of a polytope. https://doi.org/10.1007/s13324-018-0226-8

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