Radon Numbers Grow Linearly

Dömötör Pálvölgyi

doi:10.1007/s00454-021-00331-2

Abstract

Define the k-th Radon number r_k of a convexity space as the smallest number (if it exists) for which any set of r_k points can be partitioned into k parts whose convex hulls intersect. Combining the recent abstract fractional Helly theorem of Holmsen and Lee with earlier methods of Bukh, we prove that r_k grows linearly, i.e., r_kle c(r_2)cdot k.

Highlights

Define a convexity space as a pair (X, C), where X is any set of points and C, the collection of convex sets, is any family over X that contains ∅, X, and is closed under intersection and under union of nested sets
The convex hull, conv S, of some point set S ⊂ X is defined as the intersection of all convex sets containing S, i.e., conv S = {C ∈ C | S ⊂ C}; since C is closed under intersection, conv S is the minimal convex set containing C
A much investigated parameter is the Radon number rk, which is defined as the smallest number for which any set of rk points can be partitioned into k parts whose convex hulls intersect in a common point

Summary

Introduction

Define a convexity space as a pair (X , C), where X is any set of points and C, the collection of convex sets, is any family over X that contains ∅, X , and is closed under (arbitrary) intersection and under (arbitrary) union of nested sets.

Results

Conclusion