Strong maximum principle for generalized solutions to equations of the Monge-Ampère type

In this paper, we investigate the strong maximum principle for generalized solutions of Monge-Ampère type equations. We prove that the strong maximum principle holds at points where the function is strictly convex but not necessarily $C^{1,1}$ smooth, and show that it fails at non-strictly convex points. The results we obtain can be applied to various Minkowski type problems in convex geometry by the virtue of the Gauss image map.