Sphere Packings pp 199-217 | Cite as
Finite Sphere Packings
Chapter
Abstract
How can one pack m unit spheres in E n such that the volume or the surface area of their convex hull is minimal?
Let m be a positive integer, and let u be a unit vector in E n . Then, we define
, it can be regarded as a local density of Sn + Lm in Sm,n. By routine computation it follows that
Based on this observation, in 1975 L. Fejes Tóth [10] made the following conjecture about the volume case of the above problem.
$$
\begin{gathered}
{\mathbf{ }}L_m = \left\{ {2i{\text{u}}:i = 0,1,...,m - 1} \right\} \hfill \\
{\text{and}} \hfill \\
{\mathbf{ }}S_{m,n} = {\text{conv}}\left\{ {S_n + L_m } \right\}. \hfill \\
{\text{Clearly}},{\mathbf{ }}S_n + L_m {\mathbf{ }}{\text{is a packing in}}{\mathbf{ }}S_{m,n} .{\mathbf{ }}{\text{Writing}} \hfill \\
{\mathbf{ }}\mu _m \left( {S_n } \right) = \frac{{mv\left( {S_n } \right)}}
{{v\left( {S_{m,n} } \right)}}, \hfill \\
\hfill \\
\end{gathered}
$$
$$
\mu _m \left( {S_n } \right) = \frac{{m\omega }}
{{\omega _n + 2\left( {m - 1} \right)\omega _{n - 1} }} \gg \delta \left( {S_n } \right).
$$
Keywords
Unit Vector Convex Hull Unit Sphere Convex Body Isoperimetric Inequality
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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© Springer-Verlag New York, Inc. 1999