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Bounded ratios for Lorentzian matrices

Daoji Huang, June Huh, Daniel Soskin, Botong Wang

TL;DR

The paper addresses multiplicative inequalities for Lorentzian matrices by introducing bounded ratios BR(X) and showing that BR(\mathbb{L}_n^+) is the dual of the cut cone, connecting Lorentzian geometry with metric-geometry facets. It identifies a new pentagonal inequality at n≥5, proves BR(\mathbb{L}_5^+) is generated by pentagonal and triangular ratios, and provides explicit constants and extremal-ray counts; it also gives a complete description of BR on the n=3 case with an entropy-like formula for the optimal constants. The work leverages links between Lorentzian polynomials, matroid representations over tracts, and Gromov’s δ-hyperbolic spaces to place BR(\mathbb{L}_n^+) inside the framework of tree metrics and the cut cone, yielding duality results and structural insights. In addition, it establishes rank-2 Lorentzian matrices as sufficient to capture BR(\mathbb{L}_n^+), proposes sub-free conjectures for integral bounded ratios, and verifies them for small n, highlighting both geometric and combinatorial richness with potential implications for Brunn–Minkowski-type inequalities and related areas.

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.

Bounded ratios for Lorentzian matrices

TL;DR

The paper addresses multiplicative inequalities for Lorentzian matrices by introducing bounded ratios BR(X) and showing that BR(\mathbb{L}_n^+) is the dual of the cut cone, connecting Lorentzian geometry with metric-geometry facets. It identifies a new pentagonal inequality at n≥5, proves BR(\mathbb{L}_5^+) is generated by pentagonal and triangular ratios, and provides explicit constants and extremal-ray counts; it also gives a complete description of BR on the n=3 case with an entropy-like formula for the optimal constants. The work leverages links between Lorentzian polynomials, matroid representations over tracts, and Gromov’s δ-hyperbolic spaces to place BR(\mathbb{L}_n^+) inside the framework of tree metrics and the cut cone, yielding duality results and structural insights. In addition, it establishes rank-2 Lorentzian matrices as sufficient to capture BR(\mathbb{L}_n^+), proposes sub-free conjectures for integral bounded ratios, and verifies them for small n, highlighting both geometric and combinatorial richness with potential implications for Brunn–Minkowski-type inequalities and related areas.

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.

Paper Structure

This paper contains 5 sections, 18 theorems, 109 equations, 1 figure.

Table of Contents

  1. introduction
  2. The pentagonal inequality
  3. Lorentzian matrices and $\delta$-hyperbolic spaces
  4. Optimal bounding constants for $\mathbb{L}_3^+$
  5. Rank 2 Lorentzian matrices

Key Result

Theorem A

The pentagonal inequality holds for any $(p_{ij}) \in \mathbb{L}_n$: The constant $4$ is optimal when all five indices are distinct, and

Figures (1)

  • Figure 1: Optimal bounding constants for an equilateral triangle cross section of $\underline{\mathrm{BR}}(\mathbb{L}_3^+)$. The midpoints of the three edges correspond to the Alexandrov--Fenchel type inequalities. Inside the inscribed circle, the optimal bounding constant is $1$. Outside of the inscribed circle, the optimal bounding constants are given by an entropy-like function.

Theorems & Definitions (43)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Example 1.4
  • Theorem A
  • Definition 1.5
  • Theorem B
  • Corollary 1.6
  • Definition 1.7
  • Theorem C
  • ...and 33 more