## Abstract

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 R^{n} 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.

Original language | American English |
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Pages (from-to) | 531-550 |

Number of pages | 20 |

Journal | Discrete and Computational Geometry |

Volume | 27 |

Issue number | 4 |

DOIs | |

State | Published - Jun 2002 |

Externally published | Yes |