Finite Lattice Packings and the Wull{shape
نویسنده
چکیده
We consider nite lattice packings C n + K of n copies of some centrally symmetric convex body K in E d for large n. Assume that C n is a subset of a lattice , and % is at least the covering radius; namely, + %K covers the space. The parametric density (C n ; %) is deened by (C n ; %) = n V (K)=V (conv C n + %K). We show that if (C n ; %) is minimal for n large then the shape of conv C n is approximately given by Wull's condition, well-known from crystallography. Thus maximizing parametric density is equivalent to optimizing a certain Gibbs{Curie energy. We also prove that in case of lattice packings of K (allowing any packing lattice), for large n the optimal shape with respect to the parametric density is approximately a Wull{shape associated to some densest packing lattice of K.
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