Quantitative Helly-type theorems via sparse approximation

TL;DR

This paper establishes new bounds for quantitative Helly-type theorems using sparse approximation techniques, improving volume and diameter bounds for intersections of convex bodies in high-dimensional spaces.

Contribution

It introduces a sparse approximation result for polytopes in John's position and applies it to enhance bounds in quantitative Helly-type theorems for volume and diameter.

Findings

Bound for polytope approximation with at most 2d vertices.

Improved volume bound for intersection of convex bodies.

Enhanced diameter bound for intersection of convex bodies.

Abstract

We prove the following sparse approximation result for polytopes. Assume that $Q$ is a polytope in John's position. Then there exist at most $2d$ vertices of $Q$ whose convex hull $Q'$ satisfies $Q \subseteq - 2d^2 \, Q'$. As a consequence, we retrieve the best bound for the quantitative Helly-type result for the volume, achieved by Brazitikos, and improve on the strongest bound for the quantitative Helly-type theorem for the diameter, shown by Ivanov and Nasz\'odi: We prove that given a finite family $\mathcal{F}$ of convex bodies in $\mathbb{R}^d$ with intersection $K$, we may select at most $2 d$ members of $\mathcal{F}$ such that their intersection has volume at most $(c d)^{3d /2} \,\mathrm{vol}\, K$, and it has diameter at most $2 d^2 \,\mathrm{diam} \,K$, for some absolute constant $c>0$.