Yoav Kallus, Fedor Nazarov
It was conjectured by Ulam that the ball has the lowest optimal lattice packing density out of all convex, origin-symmetric three-dimensional solids. We affirm a local version of this conjecture: the ball has a lower optimal lattice packing than any body of sufficiently small asphericity in three dimensions. We also show that in dimensions 4, 5, 6, 7, 8, and 24 there are bodies of arbitrarily small asphericity that pack worse than balls.
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http://arxiv.org/abs/1212.2551
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