The spherical conjecture in Minkowski geometry
In 1991 Croft, Falconer and Guy wrote ``\dots packing \(m\) disks or balls into a convex body so that the perimeter or surface area, or alternatively the diameter is minimum, it seems that the optimum shape is roughly circular or spherical if \(m\) is large.'' This ``spherical conjecture'' was independently verified by Böröczky and Zong in 1994. Both prove that the conjecture is true in a more general setting: whenever a quermassintegral different from the volume is minimum. In this note the author shows that the ``spherical conjecture'' is also valid in Minkowski geometry. In particular he proves that the shapes of convex bodies containing \(m\) translates of a convex body \(K\) so that their Minkowskian surface area is minimum tend to a convex body \(L\) which depends on the chosen Minkowskian space.
scientific article; zbMATH DE number 2130541