#### 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