Helly numbers for Quantitative Helly-type results

TL;DR

This paper establishes new quantitative Helly-type theorems with optimal Helly numbers for convex bodies and logarithmically concave functions, including colorful variants, advancing the understanding of intersection properties in convex geometry.

Contribution

The paper introduces three new Helly-type theorems with optimal Helly numbers for diameter and pointwise minima, including colorful versions, in convex geometry.

Findings

Optimal Helly number 2d for diameter intersection

Optimal Helly number 2d+1 for logarithmically concave functions

Colorful Helly-type theorem with Helly number 3d+1

Abstract

We obtain three Helly-type results. First, we establish a Quantitative Colorful Helly-type theorem with the optimal Helly number \(2d\) concerning the diameter of the intersection of a family of convex bodies. Second, we prove a Quantitative Helly-type theorem with the optimal Helly number \(2d+1\) for the pointwise minimum of logarithmically concave functions. Finally, we present a colorful version of the latter result with Helly number (number of color classes) \(3d+1\); however, we have no reason to believe that this bound is sharp.