Sphere packing in a sphere - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T06:39:02Z http://mathoverflow.net/feeds/question/52339 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52339/sphere-packing-in-a-sphere Sphere packing in a sphere henryreed 2011-01-17T18:15:30Z 2011-01-18T18:16:40Z <p>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 $S_R^d$?</p> http://mathoverflow.net/questions/52339/sphere-packing-in-a-sphere/52421#52421 Answer by ndkrempel for Sphere packing in a sphere ndkrempel 2011-01-18T18:16:40Z 2011-01-18T18:16:40Z <p>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:</p> <ul> <li>Gritzmann and Wills, "Finite packing and covering", in Gruber and Wills, "Handbook of Convex Geometry"</li> <li>Wills, "Finite sphere packings and sphere coverings"</li> <li>Wills, "Finite sphere packings and the methods of Blichfeldt and Rankin"</li> <li>Fejes Toth, "Packing and covering", in Goodman and O'Rourke, "Handbook of Discrete and Computational Geometry"</li> <li>Nuermela and Ostergard, "Dense packings of congruent circles in a circle"</li> <li>Melissen, "Densest packing of eleven congruent circles in a circle"</li> <li>Melissen, "Packing and Covering with Circles"</li> <li>Chow, "Penny-packings with minimal second moments"</li> <li>Fejes Toth, Gritzmann and Wills, "Finite sphere packing and sphere covering"</li> <li>Borcherds and Wills, "Finite sphere packing and critical radii"</li> </ul>