Local minimality of the truncated octahedron for the isoperimetric problem on parallelohedra

TL;DR

This paper analyzes the isoperimetric problem for Voronoi cells of 3D lattices, proving BCC as a strict local and global minimizer of the isoperimetric quotient among certain lattice configurations.

Contribution

It derives an explicit formula for the isoperimetric quotient and establishes BCC as a local and global minimizer within specific lattice families.

Findings

BCC lattice is a strict local minimizer of the isoperimetric quotient.

FCC and SC lattices are not local minimizers.

BCC lattice is the global minimizer within a family interpolating between BCC and FCC.

Abstract

We investigate the isoperimetric problem for the Voronoi cells of three-dimensional lattices. Using Selling parameters, we derive an explicit closed formula for the scale-invariant isoperimetric quotient $F$ in terms of six non-negative variables. We then analyse the local behaviour of $F$ at the most relevant lattice configurations: we prove that the body-centered cubic lattice (BCC) is a strict local minimiser of $F$ at fixed volume, whereas the face-centered cubic lattice (FCC) and the simple cubic lattice (SC) are not local minimisers. Then, we consider a family of lattices which interpolates between BCC and FCC, showing that BCC is the global minimiser of $F$ restricted to this family.