Unique Determination of Ellipsoids by Their Dual Volumes

Abstract Gusakova and Zaporozhets conjectured that ellipsoids in $\mathbb R^n$ are uniquely determined (up to an isometry) by their intrinsic volumes. Petrov and Tarasov confirmed this conjecture in $\mathbb R^3$. In this paper, we solve the dual problem in all dimensions. We show that any ellipsoid in $\mathbb R^n$ centered at the origin is uniquely determined (up to an isometry) by an $n$-tuple of its dual volumes. As an application, we give an alternative proof of the result of Petrov and Tarasov.