A priori estimates and existence of solutions to the prescribed centroaffine curvature problem

In this paper we study the prescribed centroaffine curvature problem in the Euclidean space ${\mathbb{R}}^{n+1}$. This problem is equivalent to solving a Monge–Ampère equation on the unit sphere. It corresponds to the critical case of the Blaschke–Santaló inequality. By approximation from the subcritical case, and using an obstruction condition and a blow-up analysis, we obtain sufficient conditions for the a priori estimates, and the existence of solutions up to a Lagrange multiplier.