Nondispersal and density properties of infinite packings

This article is motivated by an optimization problem arising in biology. Interpreting the egg arrangements (packings) in the brood chamber as results from an optimization process, we are led to look for packings that are at the same time the most possible dense and nondispersed. We first model this issue in terms of an elementary shape optimization problem among convex bodies, involving their inradius, diameter, and area. We then solve it completely, showing that the solutions are either particular hexagons or a symmetric 2-cap body, namely the convex hull of a disk and two points lined up with the center of the disk.