Regular polyhedra and Hajós polyhedra (Q2714354)

Given two concentric circles, find the smallest (in area) polygon which contains the inner circle and has vertices on the outer circle. \textit{J. Molnar} [Magyar Tud. Akad., Mat. Fiz. Tud. Oszt. Közl. 12, 223-263 (1962; Zbl 0142.20301)] proved that the solutions (in any plane of constant curvature) are the so-called Hajós polygons, whose sides, with a possible single exception, touch the inner circle. He raised the following problem: in 3-dimensional Euclidean space determine the convex polyhedron of minimal volume if it contains a given ball and the vertices lie in a concentric sphere. In a 3-dimensional space of constant curvature a Hajós polyhedron has congruent Hajós polygon faces each touching the inscribed ball at its circumcenter. In the 60's B. Bollobás characterized Hajós polyhedra but his result was never published. The main theorem in the paper under review describes this characterization and proves that no other Hajós polyhedra exist. But there is much more in this paper, particularly inequalities (in \(d\) dimensions) involving the distances from the center of the inner sphere to the faces of the polyhedron.