Finite Sphere Packings and Critical Radii

Abstract: The parametric density permits a joint theory for classical packings and sausage packings of convex bodies, in particular spheres. If the parameter $\varrho$ is small then linear packings (``sausages") are optimal; if $\varrho$ is large then full dimensional packings (``clusters") are optimal and the packings tend to the classical densest infinite packing as the number of bodies tend to infinity. It was conjectured, namely, the Strong Sausage Conjecture, that for sphere packings no intermediate optimal packings exist, i.e. either the minimal or the maximal possible dimension of the packing configuration is attained. The proof of this conjecture would imply a proof of Kepler's conjecture for infinite sphere packings, so even in $\E^3$ only partial results can be expected.

The critical parameter depends on the dimension and on the number of spheres, so if the parameter $\varrho$ is fixed then abrupt changes of the shape of the optimal packings (``sausage catastrophe") occur as the number of spheres grows. In this paper, we give several partial results, in particular in $\E^3$, which support the conjectures.

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