A deterministic algorithm for approximating the mixed discriminant and mixed volume, and a combinatorial corollary

We present a deterministic polynomial-time algorithm that computes the mixed discriminant of an n-tuple of positive semidefinite matrices to within an exponential multiplicative factor. To this end we extend the notion of doubly stochastic matrix scaling to a larger class of n-tuples of positive semidefinite matrices, and provide a polynomial-time algorithm for this scaling. As a corollary, we obtain a deterministic polynomial algorithm that computes the mixed volume of n convex bodies in Rⁿ to within an error which depends only on the dimension. This answers a question of Dyer, Gritzmann and Hufnagel. A "side benefit" is a generalization of Rado's theorem on the existence of a linearly independent transversal.