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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.