A geometric inequality for circle packings

TL;DR

This paper introduces a geometric inequality related to circle packings, proving it through polynomial nonnegativity techniques involving sum of squares, semidefinite programming, and symmetry reduction.

Contribution

The paper presents a novel geometric inequality for circle packings and develops a proof method using polynomial nonnegativity certificates.

Findings

The inequality holds for the considered circle packing configurations.

Sum of squares and semidefinite programming effectively verify polynomial nonnegativity.

The approach simplifies proving geometric inequalities in packing problems.

Abstract

A geometric inequality among three triangles, originating in circle packing problems, is introduced. In order to prove it, we reduce the original formulation to the nonnegativity of a particular polynomial in four real indeterminates. Techniques based on sum of squares decompositions, semidefinite programming, and symmetry reduction are then applied to provide an easily verifiable nonnegativity certificate.