When is the ball a local pessimum for covering?
Yoav Kallus
TL;DR
This paper investigates whether the sphere is the worst shape for covering n-dimensional space with lattice translates, finding it is a local pessimum in 3D but not in 4D and 5D.
Contribution
It identifies the local pessimality of the sphere for covering in 3D and demonstrates it is not in 4D and 5D, advancing understanding of covering shapes.
Findings
Sphere is a local pessimum in 3D.
Sphere is not a local pessimum in 4D and 5D.
Results depend on the dimension of the space.
Abstract
We consider the problem of identifying the worst point-symmetric shape for covering n-dimensional Euclidean space with lattice translates. Here we focus on the dimensions where the thinnest lattice covering with balls is known and ask whether the ball is a pessimum for covering in these dimensions compared to all point-symmetric convex shapes. We find that the ball is a local pessimum in 3 dimensions, but not so for 4 and 5 dimensions.
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