Area minimization of special polygons

Abstract

It is quite rare that a simple area optimization result bears somebody’s name. One of these statements, called Hajos’ Lemma, became particularly known, mainly because of its esthetic appearance and due to its application at solving the densest circle packing problem. Hajos considered a pair of concentric circles and wanted to find the minimum area polygon among those polygons which contain the smaller circle and whose vertices are outside of the larger circle. In this paper we state and prove two generalizations of Hajos’ Lemma. In the first version we allow the circles to be non concentric, in the second version we consider disc polygons instead of usual polygons.