Uniqueness results for the Minkowski problem extended to hedgehogs

The classical Minkowski problem has a natural extension to hedgehogs, that is to Minkowski differences of closed convex hypersurfaces. This extended Minkowski problem is much more difficult since it essentially boils down to the question of solutions of certain Monge-Ampère equations of mixed type on the unit sphere \mathbbSⁿ \mathbb{S}^n of ℝ ⁿ⁺¹. In this paper, we mainly consider the uniqueness question and give first results.