A deterministic polynomial-time algorithm for approximating mixed discriminant and mixed volume

We present a deterministic polynomial algorithm that computes the mixed discriminant of an n-tuple of positive semidefinite matrices to within a multiplicative factor of e. 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. We obtain tight upper and lower bounds on the mixed discriminant of doubly stochasic n-tuples, proving a conjecture of Bapat, and generalizing the van der Waerden Falikman Egorychev theorem. As a corollary, we obtain a deterministic polynomial algorithm that computes the mixed volume of n convex bodies in R to within a multiplicative factor of n. This answers a question of Dyer, Gritzmann and Hufnagel.

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