An isoperimetric problem for three-dimensional parallelohedra

Zsolt Lángi

Vol. 316 (2022), No. 1, 169–181

DOI: 10.2140/pjm.2022.316.169

Abstract

The aim of this note is to investigate isoperimetric-type problems for $3$-dimensional parallelohedra; that is, for convex polyhedra whose translates tile the $3$-dimensional Euclidean space. Our main result states that among $3$-dimensional parallelohedra with unit volume, the one with minimal mean width is the regular truncated octahedron.

Keywords

parallelohedra, zonotopes, discrete isoperimetric problems, tiling, Kepler's conjecture, honeycomb conjecture, Kelvin's conjecture

Mathematical Subject Classification

Primary: 52B60

Secondary: 52A40, 52C22

Milestones

Received: 24 February 2021

Revised: 7 October 2021

Accepted: 5 November 2021

Published: 26 February 2022