Let $S_a^d$ be the $(d-1)$-dimensional sphere of radius $a$ in $\mathbb{R}^d$. Let $r>0$ be a constant and $R=\nu r$ where $\nu>1$ (some constant). Are there any known upper bounds on the number of disjoint $S_r^d$ that can be completely contained in $S_R^d$?

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    $\begingroup$ Divide the volumes and take the ceiling to obtain a trivial upper bound. $\endgroup$ Jan 17, 2011 at 18:44
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    $\begingroup$ An upper bound on the density is given by Boroczky: ams.org/mathscinet-getitem?mr=512399 $\endgroup$
    – Ian Agol
    Jan 17, 2011 at 21:03

1 Answer 1


Although the book "Sphere packings, lattices and groups" by Conway and Sloane deals mostly with infinite packings, it has an extensive bibliography that gives some references for the problem you're interested in too. Here are some entries that looked relevant, although I haven't read them:

  • Gritzmann and Wills, "Finite packing and covering", in Gruber and Wills, "Handbook of Convex Geometry"
  • Wills, "Finite sphere packings and sphere coverings"
  • Wills, "Finite sphere packings and the methods of Blichfeldt and Rankin"
  • Fejes Toth, "Packing and covering", in Goodman and O'Rourke, "Handbook of Discrete and Computational Geometry"
  • Nuermela and Ostergard, "Dense packings of congruent circles in a circle"
  • Melissen, "Densest packing of eleven congruent circles in a circle"
  • Melissen, "Packing and Covering with Circles"
  • Chow, "Penny-packings with minimal second moments"
  • Fejes Toth, Gritzmann and Wills, "Finite sphere packing and sphere covering"
  • Borcherds and Wills, "Finite sphere packing and critical radii"
  • $\begingroup$ That looks good. I'll have a good look at these! Thanks! $\endgroup$
    – alext87
    Jan 18, 2011 at 19:28

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