  In this paper, we investigate the strong maximum principle for generalized
solutions of Monge-Amp\`ere 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.
