No weak epsilon nets for lines and convex sets in space

TL;DR

This paper proves that for lines and convex sets in higher-dimensional space, there are no small, constant-size weak epsilon nets that can intersect all such sets containing a significant fraction of points.

Contribution

It establishes a fundamental limitation by proving the non-existence of small weak epsilon nets for lines and convex sets in any fixed dimension.

Findings

No weak epsilon nets of constant size exist for lines in space.

No weak epsilon nets of constant size exist for convex sets in space.

Results hold in arbitrary fixed dimensions.

Abstract

We prove that there exist no weak $\varepsilon$-nets of constant size for lines and convex sets in $\mathbb{R}^d$.