
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 , we establish the existence of a constant and of a subset of such that the following holds: for every convex set of volume at least and contains at most points at distance at most from the origin, for every large .
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