The 3-ball is a local pessimum for packing
arXiv:1212.2551 · doi:10.1016/j.aim.2014.07.015
Abstract
It was conjectured by Ulam that the ball has the lowest optimal packing fraction out of all convex, three-dimensional solids. Here we prove that any origin-symmetric convex solid of sufficiently small asphericity can be packed at a higher efficiency than balls. We also show that in dimensions 4, 5, 6, 7, 8, and 24 there are origin-symmetric convex bodies of arbitrarily small asphericity that cannot be packed using a lattice as efficiently as balls can be.
Final submitted manuscript
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