Introduction to the combinatorial atlas

TL;DR

This paper presents elementary proofs of the strong Mason conjecture and the Alexandrov-Fenchel inequality using combinatorial atlas technology, and explores their relationship with Lorentzian polynomials.

Contribution

It introduces a new combinatorial atlas framework and applies it to prove significant inequalities in combinatorics and geometry.

Findings

Elementary proofs of the strong Mason conjecture and Alexandrov-Fenchel inequality

Establishment of a formal link between combinatorial atlases and Lorentzian polynomials

Demonstration of the versatility of combinatorial atlas technology

Abstract

We give elementary self-contained proofs of the strong Mason conjecture recently proved by Anari at. al. (arXiv:1811.01600) and Br\"and\'en--Huh (arXiv:1902.03719), and of the classical Alexandrov--Fenchel inequality. Both proofs use the combinatorial atlas technology recently introduced by the authors (arXiv:2110.10740). We also give a formal relationship between combinatorial atlases and Lorentzian polynomials.