The extremals of Minkowski’s quadratic inequality

In a seminal paper “Volumen und Oberfläche” (1903), Minkowski introduced the basic notion of mixed volumes and corresponding inequalities that lie at heart convex geometry. The fundamental importance characterizing extremals these was already emphasized by himself, but has to date only been resolved in special cases. this paper, we completely settle Minkowski’s quadratic inequality, confirming conjecture R. Schneider. Our proof is based on representation arbitrary bodies as Dirichlet forms associated certain highly degenerate elliptic operators. A key ingredient quantitative rigidity property