Elliptic brunn-minkowski theory: Generalizations of the Brunn-Minkowski inequalities.

摘要

The classical Brunn-Minkowski theory for matrices studies a set of inequalities for positive definite symmetric matrices using determinants and mixed determinants. The present work expands upon some of those inequalities in two directions. 1. Inequalities are taken out of the restricted realm of positive definite symmetric matrices and 2. The inequalities' uses of determinants and mixed determinants are expanded to more generalized notions of kth elementary symmetric functions and those functions' completely polarized forms, using Garding's theory of hyperbolic polynomials.