Universal convex coverings

TL;DR

This paper proves the existence of universal convex coverings in all dimensions, ensuring large convex sets can be covered by a fixed set with controlled point density, revealing new geometric covering properties.

Contribution

It introduces a universal set that covers all large convex sets in any dimension with bounded point density, advancing understanding of convex covering problems.

Findings

Existence of universal convex covering sets in all dimensions.

Bounded point density of the covering set relative to radius.

Applicable to convex sets of sufficiently large volume.

Abstract

In every dimension $d\ge1$, we establish the existence of a constant $v_d>0$ and of a subset $\mathcal U_d$ of $\mathbb R^d$ such that the following holds: $\mathcal C+\mathcal U_d=\mathbb R^d$ for every convex set $\mathcal C\subset \mathbb R^d$ of volume at least $v_d$ and $\mathcal U_d$ contains at most $\log(r)^{d-1}r^d$ points at distance at most $r$ from the origin, for every large $r$.