Sphere packings revisited

In this paper 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.

[1]  Károly Bezdek,et al.  On the Maximum Number of Touching Pairs in a Finite Packing of Translates of a Convex Body , 2002, J. Comb. Theory, Ser. A.

[2]  K. Ball.  A lower bound for the optimal density of lattice packings , 1992 .

[3]  Thomas C. Hales.  Sphere Packings, II , 1997, Discret. Comput. Geom..

[4]  Circle packings into convex domains of the Euclidean and hyperbolic plane and the sphere , 1986 .

[5]  Károly Bezdek,et al.  Finding the Best Face on a Voronoi Polyhedron – The Strong Dodecahedral Conjecture Revisited , 2005 .

[6]  Douglas J. Muder,et al.  Putting the best face on a Voronoi polyhedron , 1988 .

[7]  C. A. Rogers,et al.  Packing and Covering , 1964 .

[8]  H. Hadwiger,et al.  Über Treffanzahlen bei translationsgleichen Eikörpern , 1957 .

[9]  Robert Connelly,et al.  Finite and Uniform Stability of Sphere Packings , 1998, Discret. Comput. Geom..

[10]  L. Danzer,et al.  Über zwei Probleme bezüglich konvexer Körper von P. Erdös und von V. L. Klee , 1962 .

[11]  W. Hsiang.  ON THE SPHERE PACKING PROBLEM AND THE PROOF OF KEPLER'S CONJECTURE , 1993 .

[12]  L. Bowen.  Circle packing in the hyperbolic plane. , 2000 .

[13]  D. Weaire,et al.  A counter-example to Kelvin's conjecture on minimal surfaces , 1994 .

[14]  O. Musin.  The kissing number in four dimensions , 2003, math/0309430.

[15]  Ulrich Betke,et al.  Sausages are good packings , 1995, Discret. Comput. Geom..

[16]  Beniamino Segre,et al.  On the Densest Packing of Circles , 1944 .

[17]  C. Petty,et al.  Equilateral sets in Minkowski spaces , 1971 .

[18]  Peter Braß,et al.  Erds Distance Problems in Normed Spaces , 1996, Comput. Geom..

[19]  K. Böröczky.  Über die Newtonsche Zahl regulärer Vielecke , 1971 .

[20]  W. Fischer,et al.  Sphere Packings, Lattices and Groups , 1990 .

[21]  Thomas C. Hales,et al.  A Proof of the Dodecahedral Conjecture , 1998, math/9811079.

[22]  Charles Radin,et al.  Densest Packing of Equal Spheres in Hyperbolic Space , 2002, Discret. Comput. Geom..

[23]  K. Bezdek,et al.  On k+-neighbour Packings and One-sided Hadwiger Configurations , 2003 .

[24]  B. L. Waerden,et al.  Das Problem der dreizehn Kugeln , 1952 .

[25]  Eiichi Bannai,et al.  Uniqueness of Certain Spherical Codes , 1981, Canadian Journal of Mathematics.

[26]  I. Talata,et al.  The Translative Kissing Number of Tetrahedra Is 18 , 1999, Discret. Comput. Geom..

[27]  A. Wyner.  Capabilities of bounded discrepancy decoding , 1965 .

[28]  Kurt M. Anstreicher.  The Thirteen Spheres: A New Proof , 2004, Discret. Comput. Geom..

[29]  N. J. A. Sloane,et al.  On Kissing Numbers in Dimensions 32 to 128 , 1998, Electron. J. Comb..

[30]  A stability property of the densest circle packing , 1988 .

[31]  David G. Larman,et al.  On the Kissing Numbers of Some Special Convex Bodies , 1999, Discret. Comput. Geom..

[32]  L. Lovász,et al.  Remarks on a theorem of Redei , 1981 .

[33]  Thomas C. Hales,et al.  An overview of the Kepler conjecture , 1998 .

[34]  D. Hilbert.  Mathematical Problems , 2019, Mathematics: People · Problems · Results.

[35]  M. Henk,et al.  Finite and infinite packings. , 1994 .

[36]  J. H. Lindsey,et al.  Sphere packing in R 3 , 1986 .

[37]  W. Hsiang.  Least Action Principle of Crystal Formation of Dense Packing Type and Kepler's Conjecture , 2002 .

[38]  John Leech,et al.  The Problem of the Thirteen Spheres , 1956, The Mathematical Gazette.

[39]  Peter Braß.  On equilateral simplices in normed spaces , 1997 .

[40]  Ludwig Danzer,et al.  Finite point-sets on S2 with minimum distance as large as possible , 1986, Discret. Math..

[41]  Florian Pfender,et al.  Kissing numbers, sphere packings, and some unexpected proofs , 2004 .

[42]  K. Böröczky.  Packing of spheres in spaces of constant curvature , 1978 .

[43]  Douglas J. Muder,et al.  A new bound on the local density of sphere packings , 1993, Discret. Comput. Geom..

[44]  J. Linhart.  Die Newtonsche Zahl von regelmässigen Fünfecken , 1973 .

[45]  Henry Cohn,et al.  New upper bounds on sphere packings I , 2001, math/0110009.

[46]  T. Hales.  The Kepler conjecture , 1998, math/9811078.

[47]  C. A. Rogers.  The Packing of Equal Spheres , 1958 .

[48]  O. Musin.  The problem of the twenty-five spheres , 2003 .

[49]  L. Fejes.  Über die dichteste Kugellagerung , 1942 .

[50]  István Talata,et al.  Exponential Lower Bound for the Translative Kissing Numbers of d -Dimensional Convex Bodies , 1998, Discret. Comput. Geom..

[51]  Samuel P. Ferguson.  Sphere packings V , 1998 .

[52]  Ulrich Betke,et al.  Finite Packings of Spheres , 1998, Discret. Comput. Geom..

[53]  K. Böröczky.  The Newton-Gregory Problem Revisited , 2003 .

[54]  H. P. F. Swinnerton-Dyer,et al.  Extremal lattices of convex bodies , 1953, Mathematical Proceedings of the Cambridge Philosophical Society.

[55]  Karoly Bezdek.  On a stronger form of Roger's lemma and the minimum surface area of Voronoi cells in unit ball packings , 2000 .

[56]  N. J. A. Sloane,et al.  New Bounds on the Number of Unit Spheres That Can Touch a Unit Sphere in n Dimensions , 1979, J. Comb. Theory, Ser. A.

[57]  Karoly Bezdek.  Improving Rogers’ Upper Bound for the Density of Unit Ball Packings via Estimating the Surface Area of Voronoi Cells from Below in Euclidean \sl d -Space for All \sl d ≥ \bf 8 , 2002, Discret. Comput. Geom..

[58]  Thomas C. Hales.  Sphere packings, I , 1997, Discret. Comput. Geom..