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.

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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