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Conic optimization for extremal geometry

Frank Vallentin

TL;DR

The paper develops and applies conic-optimization frameworks to four geometric packing problems: kissing numbers on spheres, sphere packing in Euclidean space, and measurable one- and π/2-avoiding sets. It builds a cohesive pipeline—modeling via independence graphs, formulating relaxations with Lovász theta, completely positive cones, and Lasserre hierarchies, and exploiting symmetry through harmonic analysis—to derive progressively tighter, certifiable bounds. Key contributions include finite and infinite-graph extensions, a detailed symmetry-aware computational approach, and state-of-the-art bounds for the problems (e.g., las$_2$ bounds, three-point bounds) along with rigorous verification strategies. The work lays a foundation for automatic, high-precision proofs in extremal geometry and paves the way toward potential full-resolution results given adequate computational resources.

Abstract

The aim of this paper is to highlight recent progress in using conic optimization methods to study geometric packing problems. We will look at four geometric packing problems of different kinds: two on the unit sphere -- the kissing number problem and measurable $π/2$-avoiding sets -- and two in Euclidean space -- the sphere packing problem and measurable one-avoiding sets.

Conic optimization for extremal geometry

TL;DR

The paper develops and applies conic-optimization frameworks to four geometric packing problems: kissing numbers on spheres, sphere packing in Euclidean space, and measurable one- and π/2-avoiding sets. It builds a cohesive pipeline—modeling via independence graphs, formulating relaxations with Lovász theta, completely positive cones, and Lasserre hierarchies, and exploiting symmetry through harmonic analysis—to derive progressively tighter, certifiable bounds. Key contributions include finite and infinite-graph extensions, a detailed symmetry-aware computational approach, and state-of-the-art bounds for the problems (e.g., las bounds, three-point bounds) along with rigorous verification strategies. The work lays a foundation for automatic, high-precision proofs in extremal geometry and paves the way toward potential full-resolution results given adequate computational resources.

Abstract

The aim of this paper is to highlight recent progress in using conic optimization methods to study geometric packing problems. We will look at four geometric packing problems of different kinds: two on the unit sphere -- the kissing number problem and measurable -avoiding sets -- and two in Euclidean space -- the sphere packing problem and measurable one-avoiding sets.

Paper Structure

This paper contains 18 sections, 4 theorems, 46 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Automatic reasoning for extremal problems in discrete geometry
  3. Some extremal problems in discrete geometry
  4. Structure of the remainder of the paper
  5. Modeling geometric packing problems as independence numbers of graphs
  6. Formulations and hierarchies of relaxations for the independence number
  7. The Lovász theta number
  8. A completely positive formulation
  9. A semidefinite programming hierarchy
  10. Computations: Symmetry reduction and rigorous verifications
  11. Exploiting symmetry via harmonic analysis
  12. Computation and verification of bounds
  13. Results and conclusion
  14. The kissing number problem
  15. The sphere packing problem
  16. ...and 3 more sections

Key Result

Theorem 3.1

Let $G = (V, E)$ be a finite graph. Then,

Theorems & Definitions (4)

  • Theorem 3.1
  • Theorem 3.2: Theorem 6.3 and Theorem 7.3 in DeCorteOV2022
  • Theorem 3.3
  • Theorem 4.1