Bounded ratios for Lorentzian matrices

TL;DR

This paper investigates multiplicative inequalities among entries of Lorentzian matrices, generalizing classical inequalities, and characterizes the structure of bounded ratios, revealing connections to metric geometry and graph theory.

Contribution

It identifies the cone of all bounded ratios with the dual of the cut cone and analyzes specific ratios like the pentagonal ratio.

Findings

The cone of bounded ratios is the dual of the cut cone.

Explicit bounds are obtained for Lorentzian matrices of size three.

Conjecture that all normalized bounded ratios are at most 2.

Abstract

We study multiplicative inequalities among entries of Lorentzian matrices, referred to as bounded ratios. These inequalities can be viewed as generalizations of the classical Alexandrov--Fenchel inequalities for mixed volumes. Our main structural result identifies the cone of all bounded ratios on Lorentzian matrices with the dual of the cut cone, a finitely generated integral polyhedral cone extensively studied in metric geometry and graph theory. We examine in detail the pentagonal ratio, which first appears for Lorentzian matrices of size at least five. For Lorentzian matrices of size three, we determine the optimal bounding constants across the entire cone of bounded ratios, obtaining an explicit entropy-like formula. We conjecture that any normalized bounded ratio is, in fact, bounded by 2.