Zsolt Lángi: An isoperimetric problem for three-dimensional parallelohedra

Abstract: Convex polyhedra whose translates tile the three-dimensional Euclidean space are called three-dimensional parallelohedra. Three-dimensional parallelohedra are among the best known convex polyhedra outside mathematics, their best known examples are the cube, the regular rhombic dodecahedron and the regular truncated octahedron. Despite this fact, in the presenter's knowledge, no isoperimetric problem has been proved for them in the literature. The aim of this note is to investigate isoperimetric-type problems for three-dimensional parallelohedra. Our main result states that among three-dimensional parallelohedra with unit volume the one with minimal mean width is the regular truncated octahedron. If time permits, in addition, we establish a connection between the edge lengths of three-dimensional parallelohedra and the edge densities of the translative mosaics generated by them, and use our method to prove that among translative, convex mosaics generated by a parallelohedron with a given volume, the one with minimal edge density is the face-to-face mosaic generated by cubes.