The 3-ball is a local pessimum for packing |
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Authors: | Yoav Kallus |
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Affiliation: | Center for Theoretical Science, Princeton University, Princeton, NJ 08544, United States |
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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. |
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Keywords: | Convex body Spherical harmonics Packing Sphere packing Lattice Pessimum |
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