Abstract
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.
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