The 3-ball is a local pessimum for packing

•We look for the worst shapes for dense lattice packing of convex bodies.•Sufficiently spherical, symmetric, convex solids pack more efficiently than 3-balls.•There are 4–8- and 24-dimensional bodies that cannot pack as well as balls.•In 3 dimensions, but not in these other dimensions, the ball is locally pessimal.

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.