Lagarias, J. C. Bounds for local density of sphere packings and the Kepler conjecture. (English) Zbl 1005.52011 Discrete Comput. Geom. 27, No. 2, 165-193 (2002). 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. Reviewer: William Moser (Montréal) Cited in 1 ReviewCited in 5 Documents MSC: 52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry) Keywords:sphere packing; Kepler conjecture; Hales-Ferguson PDFBibTeX XMLCite \textit{J. C. Lagarias}, Discrete Comput. Geom. 27, No. 2, 165--193 (2002; Zbl 1005.52011) Full Text: DOI arXiv