An isoperimetric problem for three-dimensional parallelohedra

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.

Corrected a minor error in Remark 2. 11 pages, 1 figure, 1 table