An isoperimetric problem for three-dimensional parallelohedra

TL;DR

This paper investigates isoperimetric problems for 3D parallelohedra, showing that the regular truncated octahedron minimizes mean width among unit-volume parallelohedra.

Contribution

It identifies the regular truncated octahedron as the shape with minimal mean width among unit-volume 3D parallelohedra, advancing understanding of geometric optimization.

Findings

Regular truncated octahedron has minimal mean width among unit-volume parallelohedra.

Among convex polyhedra tiling space, the truncated octahedron is optimal for mean width.

The study provides a characterization of optimal parallelohedra in isoperimetric terms.

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.