The 3-ball is a local pessimum for packing

TL;DR

This paper proves that in three dimensions, the sphere is not the worst packer among convex solids, and in higher dimensions, certain convex bodies cannot be packed more efficiently than spheres, challenging Ulam's conjecture.

Contribution

It demonstrates that small asphericity convex bodies can pack more efficiently than spheres in 3D and identifies bodies in higher dimensions that cannot surpass sphere packing efficiency.

Findings

In 3D, small asphericity convex bodies outperform spheres in packing efficiency.

In dimensions 4, 5, 6, 7, 8, and 24, some convex bodies cannot be packed more efficiently than spheres.

The 3-ball is not a local pessimum for packing in three dimensions.

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.