Mixed volumes of normal complexes

TL;DR

This paper develops the theory of mixed volumes for normal complexes, establishing conditions for Alexandrov-Fenchel inequalities, and applies these results to prove the Heron-Rota-Welsh Conjecture for matroids.

Contribution

It introduces a sufficiency condition for mixed volumes of normal complexes to satisfy Alexandrov-Fenchel inequalities and applies this to prove a major conjecture in matroid theory.

Findings

Mixed volumes of normal complexes can satisfy Alexandrov-Fenchel inequalities under certain conditions.

A new proof of the Heron-Rota-Welsh Conjecture is provided using these inequalities.

The study connects polyhedral geometry with combinatorial properties of matroids.

Abstract

Normal complexes are orthogonal truncations of polyhedral fans. In this paper, we develop the study of mixed volumes for normal complexes. Our main result is a sufficiency condition that ensures when the mixed volumes of normal complexes associated to a given fan satisfy the Alexandrov-Fenchel inequalities. By specializing to Bergman fans of matroids, we give a new proof of the Heron-Rota-Welsh Conjecture as a consequence of the Alexandrov-Fenchel inequalities for normal complexes.