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

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


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.


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 



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.


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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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