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

Annalisa Cesaroni, Matteo Novaga

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.

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

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 in terms of six non-negative variables. We then analyse the local behaviour of at the most relevant lattice configurations: we prove that the body-centered cubic lattice (BCC) is a strict local minimiser of 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 restricted to this family.

Paper Structure

This paper contains 11 sections, 5 theorems, 85 equations, 1 table.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Notation and preliminary definitions
  4. Analysis of parallelohedra
  5. The truncated octahedron
  6. The rhombic dodecahedron
  7. The cube
  8. Comparison with numerical simulations
  9. Global minimality of BCC on a family of lattices
  10. Two-parameters families of lattices
  11. The opposite orbit

Key Result

Theorem 3.1

Let $F$ be defined by eq:Fdef. Then $\boldsymbol{\rho}_{\mathrm{BCC}}$ is a strict local minimiser of $F$ on the hypersurface Equivalently, among lattices sufficiently close to BCC and with the same volume, BCC minimises the scale-invariant quotient $F$.

Theorems & Definitions (11)

  • Theorem 3.1: Local minimality of BCC at fixed volume
  • proof
  • Remark 3.2
  • Theorem 3.3: Local behaviour of FCC
  • proof
  • Theorem 3.4: Local behaviour of SC
  • proof
  • Proposition 4.1
  • proof
  • Theorem 4.2
  • ...and 1 more