Comparing the sets of volume polynomials and Lorentzian Polynomials

TL;DR

This paper investigates the relationship between volume polynomials of convex bodies and Lorentzian polynomials, providing a complete classification of when these two sets coincide based on polynomial properties.

Contribution

It offers a full classification of when volume polynomials of convex bodies are exactly the Lorentzian polynomials, expanding understanding of their structural relationship.

Findings

Identifies conditions under which volume polynomials equal Lorentzian polynomials

Provides a classification framework for these polynomial sets

Utilizes operations preserving Lorentzian property for analysis

Abstract

Given n convex bodies in the real space of dimension d, we consider the set of homogeneous polynomials of degree d in n variables that can be represented as their volume polynomial. This set is a subset of the set of Lorentzian polynomials. Using our knowledge of operations that preserve the Lorentzian property, we give a complete classification of the cases when the two sets are equal.