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

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.