This article is motivated by an optimization problem arising in Biology. Interpretingthe eggs arrangements (packings) in the brood chamber as results from an optimization process, weare led to look for packings that are at the same time the most possible dense and non-dispersed. Wefirst 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 solutionsare either particular hexagons or a symmetric 2-cap body, namely the convex hull of a disk and twopoints lined-up with the center of the disk.