Densest Packings of More than Three d -Spheres Are Nonplanar

We prove that for a densest packing of more than three d -balls, d geq 3 , where the density is measured by parametric density, the convex hull of their centers is either linear (a sausage) or at least three-dimensional. This is also true for restrictions to lattice packings. These results support the general conjecture that densest sphere packings have extreme dimensions. The proofs require a Lagrange-type theorem from number theory and Minkowski's theory of mixed volumes. Received November 27, 1998, and in revised form January 4, 1999. Online publication May 16, 2000.