  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
$(\frac{1}{n} I,...,\frac{1}{n} I)$. This result settles a conjecture posed by
R. Bapat in 1989. We consider various generalizations and applications of this
result.
