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