Positive-fraction intersection results and variations of weak epsilon-nets

TL;DR

This paper establishes conditions under which positive-fraction subfamilies of point-generated sets in Euclidean space have non-empty intersections, leading to new results on weak epsilon-nets and their topological variants.

Contribution

It introduces volumetric and structural criteria for positive-fraction intersections and extends weak epsilon-net existence results to topological settings.

Findings

Existence of positive-fraction intersecting subfamilies under certain conditions

Construction of weak epsilon-nets for point-generated families

Topological variation of weak epsilon-net existence

Abstract

Given a finite set $X$ of points in $R^n$ and a family $F$ of sets generated by the pairs of points of $X$, we determine volumetric and structural conditions for the sets that allow us to guarantee the existence of a positive-fraction subfamily $F'$ of $F$ for which the sets have non-empty intersection. This allows us to show the existence of weak epsilon-nets for these families. We also prove a topological variation of the existence of weak epsilon-nets for convex sets.