An isoperimetric problem for three-dimensional parallelohedra

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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