Van der Waerden Conjecture for Mixed Discriminants

We prove that the mixed discriminant of doubly stochastic $n$-tuples of semidefinite hermitian $n\times n$ matrices is bounded below by $\frac{n!}{n^n}$ and that this bound is uniquely attained at the $n$-tuple $(\frac{1}{n}I,\mathrm{.}\mathrm{.}\mathrm{.},\frac{1}{n}I)$. This result settles a conjecture posed by R. Bapat in 1989. We consider various generalizations and applications of this result.