The inverse moment problem for convex polytopes

TL;DR

This paper introduces a novel method for reconstructing convex polytopes from a limited set of moments, leveraging geometric formulas and algebraic techniques to recover vertices efficiently.

Contribution

It presents a new approach combining moment formulas and algebraic methods to reconstruct convex polytopes from axial moments in arbitrary directions.

Findings

Vertices can be reconstructed from O(DN) moments.

The method applies to polytopes in any dimension d.

Uses classical geometric formulas and algebraic factorization techniques.

Abstract

The goal of this paper is to present a general and novel approach for the reconstruction of any convex d-dimensional polytope P, from knowledge of its moments. In particular, we show that the vertices of an N-vertex polytope in R^d can be reconstructed from the knowledge of O(DN) axial moments (w.r.t. to an unknown polynomial measure od degree D) in d+1 distinct generic directions. Our approach is based on the collection of moment formulas due to Brion, Lawrence, Khovanskii-Pukhikov, and Barvinok that arise in the discrete geometry of polytopes, and what variously known as Prony's method, or Vandermonde factorization of finite rank Hankel matrices.