Vol. 316, No. 1, 2022

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An isoperimetric problem for three-dimensional parallelohedra

Zsolt Lángi

Vol. 316 (2022), No. 1, 169–181
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
Authors
Zsolt Lángi
Morphodynamics Research Group and Department of Geometry
Budapest University of Technology
Budapest
Hungary