Finite sphere packings and critical radii. (English) Zbl 0890.52009

Böröczky, K. jun.; Wills, J. M.

Beitr. Algebra Geom. 38, No. 2, 193-211 (1997).

\(B^d\) 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.

Reviewer: W.Moser (Montreal)

MSC:

52C17

Packing and covering in \(n\) dimensions (aspects of discrete geometry)

Keywords:

strong sausage conjecture; packings; spheres