Projections of convex polytopes to a line and higher univariate Prony systems

TL;DR

This paper investigates the pushforward of Lebesgue measure on convex polytopes to a line, revealing connections to spline densities, Prony systems, and moment varieties, with explicit criteria and algebraic structures.

Contribution

It introduces a new framework linking convex polytope projections to higher univariate Prony systems and characterizes the associated algebraic and geometric structures.

Findings

Describes the fixed-knot spline cone for the projection densities.

Provides an explicit amplitude recovery criterion.

Identifies the moment variety with a Hankel determinantal variety.

Abstract

Motivated by the inverse moment problem for convex polytopes, we study the pushforward to a line of the Lebesgue measure restricted to a convex $d$-polytope. Such pushforwards are spline densities of degree $d-1$, and their moments lead naturally to a family of ``higher'' univariate Prony systems, with the classical Prony system recovered when $d=0$. We describe the corresponding fixed-knot spline cone, give an explicit amplitude recovery criterion, record the rational generating function and recurrence satisfied by the normalized moments, and identify the directional moment variety with the Hankel determinantal variety appearing in the theory of moment varieties of measures on polytopes.