Sphere packings revisited. (English) Zbl 1091.52010
Summary: We survey most of the recent and often surprising results on packings of congruent spheres in \(d\)-dimensional spaces of constant curvature. The topics discussed are as follows:
– Hadwiger numbers of convex bodies and kissing numbers of spheres;
– touching numbers of convex bodies;
– Newton numbers of convex bodies;
– one-sided Hadwiger and kissing numbers;
– contact graphs of finite packings and the combinatorial Kepler problem;
– isoperimetric problems for Voronoi cells, the strong dodecahedral conjecture and the truncated octahedral conjecture;
– the strong Kepler conjecture;
– bounds on the density of sphere packings in higher dimensions;
– solidity and uniform stability.
Each topic is discussed in details along with some of the “most wanted” research problems.
– Hadwiger numbers of convex bodies and kissing numbers of spheres;
– touching numbers of convex bodies;
– Newton numbers of convex bodies;
– one-sided Hadwiger and kissing numbers;
– contact graphs of finite packings and the combinatorial Kepler problem;
– isoperimetric problems for Voronoi cells, the strong dodecahedral conjecture and the truncated octahedral conjecture;
– the strong Kepler conjecture;
– bounds on the density of sphere packings in higher dimensions;
– solidity and uniform stability.
Each topic is discussed in details along with some of the “most wanted” research problems.
MSC:
| 52C17 | Packing and covering in \(n\) dimensions (aspects of discrete geometry) |
Keywords:
packings of congruent spheres; \(d\)-dimensional spaces of constant curvature; Hadwiger numbers; touching numbers; Newton numbers; isoperimetric problems; Voronoi cells; dodecahedral conjecture; octahedral conjecture; Kepler conjectureSoftware:
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