On weak $\epsilon$-nets and the Radon number

TL;DR

This paper explores the relationship between the Radon number and the existence of weak epsilon-nets in convexity spaces, providing new characterizations, constructions, and bounds based on combinatorial and geometric properties.

Contribution

It introduces a characterization of weak epsilon-nets via the Radon number and develops new bounds and constructions using Helly's property, VC classes, and Kneser graph chromatic numbers.

Findings

Radon number characterizes weak nets in convexity spaces

Constructs weak nets using Helly's property and VC class metrics

Establishes lower bounds on weak net sizes based on Kneser graph chromatic number

Abstract

We show that the Radon number characterizes the existence of weak nets in separable convexity spaces (an abstraction of the euclidean notion of convexity). The construction of weak nets when the Radon number is finite is based on Helly's property and on metric properties of VC classes. The lower bound on the size of weak nets when the Radon number is large relies on the chromatic number of the Kneser graph. As an application, we prove a boosting-type result for weak $\epsilon$-nets.