Finite sphere packings and critical radii (Q1367999)

\(B^d\) denotes the unit ball in \(E^d\), \(V(K)\) the volume of the convex body \(K\), \(C_n\) a set of \(n\) points in \(E^d\) with distinct pairs at distance \(\geq 2\) so that \(C_n+ B^d\) is a packing \((+\) is the Minkowski sum). For \(\rho>0\) the parametric density of the packing is \({nV(B^d) \over V(\text{conv} C_n +\rho B^d)}\). When the parameter is small, linear packings (``sausages'') are optimal; if \(\rho\) is large, full dimensional packings (``clusters'') are optimal. The Strong Sausage Conjecture is that for sphere packings no intermediate optimal packings exist. If \(\rho\) is fixed then abrupt changes of the shape of the optimal packings (``sausage catastrophes'') occur as the number of spheres grows. In this remarkable paper there are 8 lemmas and 8 theorems (and 28 references) of substantial partial results, particularly in \(E^3\), which support the conjectures.