Since its creation by Brunn and Minkowski, what has become known as the Brunn Minkowski theory has provided powerful machinery to solve a broad variety of inverse problems with stereological data. The machinery of the Brunn Minkowski theory includes mixed volumes (of Minkowski), symmetrization techniques (such as those of Steiner and Blaschke), isoperimetric inequalities (such as the Brunn Mink...

In this paper we establish the Lp-Minkowski inequality and Lp-Aleksandrov-Fenchel type inequality for Lp-dual mixed volumes of star duality of mixed intersection bodies, respectively. As applications, we get some related results. The paper new contributions that illustrate this duality of projection and intersection bodies will be presented. M.S.C. 2000: 52A40.

We construct a probabilistic polynomial time algorithm that computes the mixed discriminant of given n positive definite n × n matrices within a 2O(n) factor. As a corollary, we show that the permanent of an n×n nonnegative matrix and the mixed volume of n ellipsoids inRn can be computed within a 2O(n) factor by probabilistic polynomial time algorithms. Since every convex body can be approximat...

In this paper we consider mixed volumes of combinations of hypersimplices. These numbers, called “mixed Eulerian numbers”, were first considered by A. Postnikov and were shown to satisfy many properties related to Eulerian numbers, Catalan numbers, binomial coefficients, etc. We give a general combinatorial interpretation for mixed Eulerian numbers and prove the above properties combinatorially...

The Brunn Minkowski theory of convex bodies and mixed volumes has provided many tools for solving problems involving projections and valuations of compact convex sets in Euclidean space. Among the most beautiful results of twentieth century convexity is Hadwiger's characterization theorem for the elementary mixed volumes (Quermassintegrals); (see [3, 5, 9]). Hadwiger's characterization leads to...