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

$$B^d$$ 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
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