On the densest packing of convex discs

Extract

We shall give a simple new proof of the following known theorem [1, 2].

Theorem. The upper density of a packing of translates of a convex disc cannot exceed the density of the densest lattice-packing of these discs.

In [] and [2] this theorem is proved for centrally symmetric discs. The general case can be reduced to this one by applying to the discs the known construction of central symmetrization. Our proof goes in a reverse way. We shall give a direct proof for a special family of asymmetric discs whose centrally symmetric images exhaust the family of centrally symmetric convex discs. Using the properties of the symmetrization, this implies the validity of the theorem for all centrally symmetric discs, and consequently for all convex discs. This procedure of going from a special case to the general one, by applying the symmetrization twice, is illustrated by the following example. The validity of the theorem for a Reuleaux triangle implies its validity for a circle, which implies its validity for any disc of constant width.