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.

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