Homothetic packings of centrally symmetric convex bodies

A centrally symmetric convex body is a convex compact set with non-empty interior that is symmetric about the origin. Of particular interest are those that are both smooth and strictly convex-known here as regular symmetric bodies-since they retain many of the useful properties of the d-dimensional Euclidean ball. We prove that for any given regular symmetric body C, a homothetic packing of copies of C with randomly chosen radii will have a (2, 2)-sparse planar contact graph. We further prove that there exists a comeagre set of centrally symmetric convex bodies C where any (2, 2)-sparse planar graph can be realised as the contact graph of a stress-free homothetic packing of C.