Vol. 316, No. 1, 2022

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 331: 1  2
Vol. 330: 1  2
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Online Archive
Volume:
Issue:
     
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Officers
 
Subscriptions
 
ISSN 1945-5844 (electronic)
ISSN 0030-8730 (print)
 
Special Issues
Author index
To appear
 
Other MSP journals
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