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Bounds for local density of sphere packings and the Kepler conjecture. (English) Zbl 1005.52011

Except for this opening sentence and the last sentence this review consists of a subset of the introduction.
The Kepler conjecture, stated by Kepler in 1611, asserts: Any packing of unit spheres in \(\mathbb{R}^3\) has upper packing density \(\pi/18\) (the density of the face-centered cubic lattice).
In recent years T. C. Hales has developed an approach for proving this based on finding a local density inequality that gives a (sharp) upper bound on the density. In 1998 he announced a proof, completed with the help of S. P. Ferguson. The proof is computer intensive, and involves checking over 5000 subproblems. It involves several new ideas which are indicated in Sections 4 and 5.
The objects of this paper are:
(i) To formulate local density inequalities for sphere packings in arbitrary dimension \(\mathbb{R}^n\), in sufficient generality to include the known candidates for optimal local inequalities in \(\mathbb{R}^3\).
(ii) To review the history of local density inequalities for three-dimensional sphere packing and the Kepler conjecture.
(iii) To give a precise statement of the local density inequality considered in the Hales-Ferguson approach.
(iv) To outline some features of the Hales-Ferguson proof.
The author succeeds admirably in his objectives.

MSC:

52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
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