  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.
