No weak epsilon nets for lines and convex sets in space

Abstract

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

Boris Bukh has informed us that our construction can be extended to 2-flats in ${\mathbb{R}}^4$. The contents of this manuscript was published as part of the paper: O. Cheong, X. Goaoc, A. F. Holmsen, Some New Results on Geometric Transversals. Discrete Comput Geom (2023)