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Computing Upper Bounds for the Packing Density of Congruent Copies of a Convex Body

  • Fernando Mário de Oliveira FilhoEmail author
  • Frank Vallentin
Chapter
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 27)

Abstract

In this paper we prove a theorem that provides an upper bound for the density of packings of congruent copies of a given convex body in \(\mathbb {R}^n\); this theorem is a generalization of the linear programming bound for sphere packings. We illustrate its use by computing an upper bound for the maximum density of packings of regular pentagons in the plane. Our computational approach is numerical and uses a combination of semidefinite programming, sums of squares, and the harmonic analysis of the Euclidean motion group. We show how, with some extra work, the bounds so obtained can be made rigorous.

Keywords

Tetrahedra packings Pentagon packings Sphere packings Lovász theta number Delsarte’s method Euclidean motion group Polynomial optimization Semidefinite programming 

1991 Mathematics Subject Classification

52C17 90C22 

Notes

Acknowledgements

We are thankful to Pier Daniele Napolitani and Claudia Addabbo from the Maurolico Project, who provided us with a transcript of Maurolico’s manuscript. In particular, Claudia Addabbo provided us with a draft of her commented Italian translation of the manuscript.

References

  1. 1.
    N.I. Akhiezer, Lectures on Integral Transforms, in Translations of Mathematical Monographs 70 (American Mathematical Society, 1988)Google Scholar
  2. 2.
    G.E. Andrews, R. Askey, R. Roy, in Special Functions, Encyclopedia of Mathematics and its Applications 71 (Cambridge University Press, Cambridge, 1999)Google Scholar
  3. 3.
    Aristotle, On the Heavens, translation by W.K.C. Guthrie (Harvard University Press, Cambridge, 2006)Google Scholar
  4. 4.
    S. Atkinson, Y. Jiao, S. Torquato, Maximally dense packings of two-dimensional convex and concave noncircular particles. Phys. Rev. E 86, 031302 (2012)Google Scholar
  5. 5.
    E. Aylward, S. Itani, P.A. Parrilo, Explicit SOS decompositions of univariate polynomial matrices and the Kalman-Yakubovich-Popov lemma, in Proceedings of the 46th IEEE Conference on Decision and Control (2007), pp. 5660–5665Google Scholar
  6. 6.
    C. Bachoc, G. Nebe, F.M. de Oliveira Filho, F. Vallentin, Lower bounds for measurable chromatic numbers. Geom. Funct. Anal. 19, 645–661 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    A. Ben-Tal, A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications (SIAM, Philadelphia, 2001)Google Scholar
  8. 8.
    A. Bezdek, W. Kuperberg, Dense packing of space with various convex solids, in Geometry — Intuitive, Discrete, and Convex, A Tribute to László Fejes Tóth, Bolyai Society Mathematical Studies, ed. by I. Bárány, K.J. Böröczky, G. Fejes Tóth, J. Pach (Springer, Berlin, 2013), pp. 66–90Google Scholar
  9. 9.
    S. Bochner, Hilbert distances and positive definite functions. Ann. Math. 42, 647–656 (1941)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    B. Borchers, CSDP, A C library for semidefinite programming. Optim. Methods Softw. 11, 613–623 (1999)Google Scholar
  11. 11.
    P. Brass, W. Moser, J. Pach, Research Problems in Discrete Geometry (Springer, Berlin, 2005)Google Scholar
  12. 12.
    B. Casselman, Can you do better? in Feature Column of the AMS, http://www.ams.org/samplings/feature-column/fc-2012-11 (2012)
  13. 13.
    E.R. Chen, M. Engel, S.C. Glotzer, Dense crystalline dimer packings of regular tetrahedra. Discrete Comput. Geom. 44, 253–280 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    M.D. Choi, T.Y. Lam, B. Reznick, Real zeros of positive semidefinite forms I. Mathematische Zeitschrift 171, 1–26 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    H. Cohn, N.D. Elkies, New upper bounds on sphere packings I. Ann. Math. 157, 689–714 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    H. Cohn, A. Kumar, Optimality and uniqueness of the Leech lattice among lattices. Ann. Math. 170, 1003–1050 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    H. Cohn, A. Kumar, S.D. Miller, D. Radchenko, and M.S. Viazovska, The sphere packing problem in dimension 24. Ann. Math. (2) 185(3), 1017–1033 (2017). arXiv:1603.06518 [math.NT]MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    H. Cohn, S.D. Miller, Some properties of optimal functions for sphere packing in dimensions 8 and 24 (2016) 23p. arXiv:1603.04759 [math.MG]
  19. 19.
    H. Cohn, Y. Zhao, Sphere packing bounds via spherical codes. Duke Math. J. 163, 1965–2002 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    J.B. Conway, A Course in Functional Analysis, Graduate Texts in Mathematics 96 (Springer, New York, 1985)zbMATHCrossRefGoogle Scholar
  21. 21.
    J.H. Conway, N.J.A. Sloane, Sphere packings, lattices and groups (Grundlehren der mathematischen Wissenschaften), vol. 290, 3rd edn. (Springer, New York, 1999)Google Scholar
  22. 22.
    J.H. Conway, S. Torquato, Packing, tiling, and covering with tetrahedra. Proc. Natl. Acad. Sci. USA 103, 10612–10617 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    E. de Klerk, F. Vallentin, On the Turing model complexity of interior point methods for semidefinite programming. SIAM J. Optim. 26(3), 1944–1961 (2016). arXiv:1507.03549 [math.OC]MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    D. de Laat, F.M. de Oliveira Filho, F. Vallentin, Upper bounds for packings of spheres of several radii. Forum Math. Sigma 2, e23 (42 pages) (2014)Google Scholar
  25. 25.
    P. Delsarte, J.M. Goethals, J.J. Seidel, Spherical codes and designs. Geom. Dedic. 6, 363–388 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    P. Delsarte, V.I. Levensthein, Association schemes and coding theory. IEEE Trans. Inf. Theory IT–44, 2477–2504 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    M. Dostert, C. Guzmán, F.M. de Oliveira Filho, F. Vallentin, New upper bounds for the density of translative packings of three-dimensional convex bodies with tetrahedral symmetry. Discrete Comput. Geom. 58, 449–481 (2017). arXiv:1510.02331 [math.MG]MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    G. Fejes Tóth, F. Fodor, V. Vígh, The packing density of the \(n\)-dimensional cross-polytope. Discrete Comput. Geom. 54, 182–194 (2015)Google Scholar
  29. 29.
    G. Fejes Tóth, W. Kuperberg, Packing and covering with convex sets, in Handbook of Convex Geometry, ed. by P.M. Gruber, J.M. Wills (North-Holland, Amsterdam, 1993), pp. 799–860Google Scholar
  30. 30.
    G.B. Folland, A Course in Abstract Harmonic Analysis (Studies in Advanced Mathematics, CRC Press, Boca Raton, 1995)Google Scholar
  31. 31.
    S. Gravel, V. Elser, Y. Kallus, Upper bound on the packing density of regular tetrahedra and octahedra. Discrete Comput. Geom. 46, 799–818 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    T.C. Hales, A proof of the Kepler conjecture. Ann. Math. 162, 1065–1185 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    T.C. Hales, M. Adams, G. Bauer, D. Tat Dang, J. Harrison, T. Le Hoang, C. Kaliszyk, V. Magron, S. McLaughlin, T. Tat Nguyen, T. Quang Nguyen, T. Nipkow, S. Obua, J. Pleso, J. Rute, A. Solovyev, A. Hoai Thi Ta, T. Nam Tran, D. Thi Trieu, J. Urban, K. Khac Vu, R. Zumkeller, A formal proof of the Kepler conjecture (2015) 21p. arXiv:1501.02155 [math.MG]
  34. 34.
    T.C. Hales, W. Kusner, Packings of regular Pentagons in the plane (2016) 26p. arXiv:1602.07220 [math.MG]
  35. 35.
    Y. Kallus, W. Kusner, The local optimality of the double lattice packing (2015) 23p. arXiv:1509.02241 [math.MG]
  36. 36.
    R.M. Karp, Reducibility among combinatorial problems, in: Complexity of Computer Computations, ed. by R.E. Miller, J.W. Thatcher. Proceedings of a symposium on the Complexity of Computer Computations, (IBM Thomas J. Watson Research Center, Yorktown Heights, Plenum Press, New York, 1972), pp. 85–103Google Scholar
  37. 37.
    J. Kepler, Vom sechseckigen Schnee (Strena seu de Nive sexangula, published in 1611), translation with introduction and notes by Dorothea Goetz, Ostwalds Klassiker der exakten Wissenschaften 273, (Akademische Verlagsgesellschaft Geest u. Portig K.-G, Leipzig, 1987)Google Scholar
  38. 38.
    G. Kuperberg, W. Kuperberg, Double-lattice packings of convex bodies in the place. Discrete Comput. Geom. 5, 389–397 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    J.C. Lagarias, C. Zong, Mysteries in packing regular tetrahedra. Notices Amer. Math. Soc. 59, 1540–1549 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    M. Laurent, Sums of squares, moment matrices and optimization, in Emerging Applications of Algebraic Geometry, IMA Volumes in Mathematics and its Applications, ed. by M. Putinar, S. Sullivant (Springer, Berlin, 2009), pp. 157–270Google Scholar
  41. 41.
    L. Lovász, On the Shannon capacity of a graph. IEEE Trans. Inf. Theory IT–25, 1–7 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    F. Maurolico, De quinque solidis, quae vulgo regularia dicuntur, quae videlicet eorum locum impleant, et quae non, contra commentatorem Aristotelis, Averroem, 1529Google Scholar
  43. 43.
    R.J. McEliece, E.R. Rodemich, H.C. Rumsey Jr., The Lovász bound and some generalizations. J. Comb. Inf. Syst. Sci. 3, 134–152 (1978)Google Scholar
  44. 44.
    C.A. Rogers, Packing and Covering (Cambridge University Press, 1964)Google Scholar
  45. 45.
    A. Schrijver, A comparison of the Delsarte and Lovász bounds. IEEE Trans. Inf. Theory IT–25, 425–429 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    W.A. Stein et al. Sage Mathematics Software (Version 4.8). The Sage Development Team (2012). http://www.sagemath.org
  47. 47.
    M. Sugiura, Unitary Representations and Harmonic Analysis: An Introduction (Kodansha Scientific Books, Tokyo, 1990)Google Scholar
  48. 48.
    M.S. Viazovska, The sphere packing problem in dimension 8. Ann. Math. (2) 185(3), 991–1015 (2017). arXiv:1603.04246 [math.NT]MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    G.N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, 1922)Google Scholar
  50. 50.
    G.M. Ziegler, Three mathematics competitions, in An Invitation to Mathematics: From Competitions to Research, ed. by D. Schleicher, M. Lackmann (Springer, Berlin, 2011), pp. 195–206CrossRefGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Fernando Mário de Oliveira Filho
    • 1
    Email author
  • Frank Vallentin
    • 2
  1. 1.Faculty of Mathematics and Computer ScienceTU DelftXE DelftThe Netherlands
  2. 2.Mathematisches Institut, Universität zu KölnKölnGermany

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