Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-09T02:55:32.249Z Has data issue: false hasContentIssue false
  • English
  • Français

Upper bounds for packings of spheres of several radii

Part of: Discrete geometry

Published online by Cambridge University Press:  23 September 2014

DAVID DE LAAT ,
FERNANDO MÁRIO DE OLIVEIRA FILHO  and
FRANK VALLENTIN
DAVID DE LAAT
Affiliation:
Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands; mail@daviddelaat.nl
FERNANDO MÁRIO DE OLIVEIRA FILHO
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, 05508-090 São Paulo, Brazil; fmario@gmail.com
FRANK VALLENTIN
Affiliation:
Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany; frank.vallentin@uni-koeln.de

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve the bounds for the classical problem of packing identical spheres.

MSC classification

Secondary: 52C17: Packing and covering in $n$ dimensions 90C22: Semidefinite programming
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e23
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2014

References

Andrews, G. E., Askey, R. and Roy, R., ‘Special functions’, Encyclopedia of Mathematics and its Applications, 71 (Cambridge University Press, Cambridge, 1999).Google Scholar
Aylward, E., Itani, S. and Parrilo, P. A., ‘Explicit SOS decompositions of univariate polynomial matrices and the Kalman–Yakubovich–Popov lemma’, Proceedings of the 46th IEEE Conference on Decision and Control (2007).Google Scholar
Bachoc, C., Nebe, G., de Oliveira Filho, F. M. and Vallentin, F., ‘Lower bounds for measurable chromatic numbers’, Geom. Funct. Anal. 19 (2009), 645–661. arXiv:0801.1059.Google Scholar
Bachoc, C. and Vallentin, F., ‘New upper bounds for kissing numbers from semidefinite programming’, J. Amer. Math. Soc. 21 (2008), 909–924. arXiv:math/0608426.Google Scholar
Bochner, S., ‘Hilbert distances and positive definite functions’, Ann. of Math. (3) 42 (1941), 647–656.Google Scholar
Bochner, S., Vorlesungen über Fouriersche Integrale (Akademische Verlagsgesellschaft, Leipzig, 1932).Google Scholar
Choi, M. D., Lam, T. Y. and Reznick, B., ‘Real zeros of positive semidefinite forms I’, Math. Z. 171 (1980), 1–26.Google Scholar
Cohn, H. and Elkies, N. D., ‘New upper bounds on sphere packings I’, Ann. of Math. (2) 157 (2003), 689–714. arXiv:math/0110009.CrossRefGoogle Scholar
Cohn, H. and Kumar, A., ‘Universally optimal distribution of points on spheres’, J. Amer. Math. Soc. 20 (2007), 99–148. arXiv:math/0607446.Google Scholar
Cohn, H. and Kumar, A., ‘Optimality and uniqueness of the Leech lattice among lattices’, Ann. of Math. (2) 170 (2009), 1003–1050. arXiv:math/0403263.Google Scholar
Courant, R. and Hilbert, D., Methods of Mathematical Physics (Interscience Publishers, 1953).Google Scholar
Delsarte, P., Goethals, J. M. and Seidel, J. J., ‘Spherical codes and designs’, Geom. Dedicata 6 (1977), 363–388.Google Scholar
Florian, A., ‘Ausfüllung der Ebene durch Kreise’, Rend. Circ. Mat. Palermo 9 (1960), 300–312.Google Scholar
Florian, A., ‘Packing of incongruent circles on the sphere’, Monatsh. Math. 133 (2001), 111–129.Google Scholar
Florian, A., ‘Remarks on my paper: packing of incongruent circles on the sphere’, Monatsh. Math. 152 (2007), 39–43.Google Scholar
Florian, A. and Heppes, A., ‘Packing circles of two different sizes on the sphere II’, Period. Math. Hungar. 39 (1999), 125–127.CrossRefGoogle Scholar
Folland, G. B., ‘A course in abstract harmonic analysis’, Studies in Advanced Mathematics (CRC Press, Boca Raton, 1995).Google Scholar
Gijswijt, D. C., ‘Matrix algebras and semidefinite programming techniques for codes’, PhD thesis, University of Amsterdam, 2005, arXiv:1007.0906.Google Scholar
Fujisawa, K., Fukuda, M., Kobayashi, K., Kojima, M., Nakata, K., Nakata, M. and Yamashita, M., SDPA (SemiDefinite Programming Algorithm) Users Manual — Version 7.0.5, Research Report B-448, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, 2008, http://sdpa.sourceforge.net.Google Scholar
Grötschel, M., Lovász, L. and Schrijver, A., ‘The ellipsoid method and its consequences in combinatorial optimization’, Combinatorica 1 (1981), 169–197.Google Scholar
Hales, T. C., ‘A proof of the Kepler conjecture’, Ann. of Math. (2) 162 (2005), 1065–1185.Google Scholar
Heppes, A., ‘Some densest two-size disc packings in the plane’, Discrete Comput. Geom. 30 (2003), 241–262.CrossRefGoogle Scholar
Heppes, A. and Kertész, G., ‘Packing circles of two different sizes on the sphere: intuitive geometry’, Bolyai Soc. Math. Stud. 6 (1997), 357–365.Google Scholar
Higham, N. J., ‘A survey of condition number estimation for triangular matrices’, SIAM Rev. 29 (1987), 575–595.CrossRefGoogle Scholar
Higham, N. J., ‘Upper bounds for the condition number of a triangular matrix’, Numerical Analysis Report No. 86, University of Manchester, Manchester, 1983.Google Scholar
Hopkins, A. B., Jiao, Y., Stillinger, F. H. and Torquato, S., ‘Phase diagram and structural diversity of the densest binary sphere packings’, Phys. Rev. Lett. 107 125501 (2011), 5pp. arXiv:1108.2210.Google Scholar
Hopkins, A. B., Stillinger, F. H. and Torquato, S., ‘Densest binary sphere packings’, Phys. Rev. E 85 021130 (2012), 19pp. arXiv:1111.4917.Google Scholar
Laurent, M., ‘Sums of squares, moment matrices and optimization over polynomials’, inEmerging Applications of Algebraic Geometry, (eds Putinar, M. and Sullivant, S.), IMA Volumes in Mathematics and its Applications, 149 (Springer, New York, 2009), 157–270.CrossRefGoogle Scholar
Levenshtein, V. I., ‘Universal bounds for codes and designs’, inHandbook of Coding Theory, Vol. I (North-Holland, Amsterdam, 1998), 499–648.Google Scholar
Löfberg, J., ‘Pre- and post-processing sums-of-squares programs in practice’, IEEE Trans. Automat. Control 54 (2009), 1007–1011.CrossRefGoogle Scholar
Löfberg, J., ‘Strictly feasible sums-of-squares solutions, post in YALMIP Wiki’, 2011.Google Scholar
Masnick, B. and Wolf, J., ‘On linear unequal error protection codes’, IEEE Trans. Inform. Theory 13 (1967), 600–607.Google Scholar
Mittelmann, H. D. and Vallentin, F., ‘High accuracy semidefinite programming bounds for kissing numbers’, Experiment. Math. 19 (2010), 174–178. arXiv:0902.1105.Google Scholar
de Oliveira Filho, F. M., ‘SDPSL: A semidefinite programming specification library’,http://www.ime.usp.br/∼fmario/sdpsl.html.Google Scholar
Rogers, C. A., ‘The packing of equal spheres’, Proc. London Math. Soc. 8 (1958), 609–620.Google Scholar
Schoenberg, I. J., ‘Positive definite functions on spheres’, Duke Math. J. 9 (1942), 96–108.Google Scholar
Stein, W. A. et al. , ‘Sage mathematics software (version 4.8), the sage development team’, 2012, http://www.sagemath.org.Google Scholar
Szegö, G., Orthogonal Polynomials, American Mathematical Society Colloquium Publications, XXIII (American Mathematical Society, Providence, 1975).Google Scholar
Terras, A., Harmonic Analysis on Symmetric Spaces and Applications I (Springer, Berlin, Heidelberg, New York, 1985).CrossRefGoogle Scholar
Torquato, S., Random Heterogeneous Materials, Microstructure and Macroscopic Properties (Springer, New York, 2002).Google Scholar
Supplementary material: File

De Laat Supplementary Material

Supplementary Material

Download De Laat Supplementary Material(File)
File 263.5 KB