A generalized rotationally symmetric case of the centroaffine Minkowski problem (Q1701864)

The paper deals with a centroaffine Minkowski problem, that is equivalent to solving the following Monge-Ampère-type equation \[ \det\big(\nabla^2H+HI\big)=\dfrac{f}{H^{n+2}}\quad \text{on}\;S^n, \] where \(f\) is a given positive function, \(H\) is the support function of a bounded convex body \(X\subset\mathbb{R}^{n+1},\) \(I\) is the unit matrix, and \(\nabla^2H=(\nabla_{ij}H)\) stands for the Hessian matrix of covariant derivatives of \(H\) with respect to an orthonormal frame over \(S^n.\) The author provides two sufficient conditions for existence of generalized rotationally symmetric solutions. The approach relies on the variational structure of the equation and blow-up analysis.