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

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.