Unique determination of ellipsoids by their dual volumes and the moment problem

TL;DR

This paper proves that in any dimension, centered ellipsoids are uniquely identified by their dual Steiner polynomials, solving a dual problem and providing an alternative proof for a known 3D case.

Contribution

It establishes the unique determination of centered ellipsoids by dual Steiner polynomials in all dimensions, extending previous 3D results and linking to moment problems.

Findings

Centered ellipsoids are uniquely determined by dual Steiner polynomials in any dimension.

The problem reduces to a moments problem, providing a new proof approach.

An alternative proof of the 3D case by Petrov and Tarasov is given.

Abstract

Gusakova and Zaporozhets conjectured that ellipsoids in $\mathbb R^n$ are uniquely determined (up to an isometry) by their Steiner polynomials. Petrov and Tarasov confirmed this conjecture in $\mathbb R^3$. In this paper we solve the dual problem. We show that any ellipsoid in $\mathbb{R}^n$ centered at the origin is uniquely determined (up to an isometry) by its dual Steiner polynomial. To prove this result we reduce it to a problem of moments. As a by-product we give an alternative proof of the result of Petrov and Tarasov.