BRILLOUIN ZONES OF INTEGER LATTICES AND THEIR PERTURBATIONS.

For a locally finite set, A ⊆ R^d, the kth Brillouin zone of a ∈ A is the region of points x ∈R^d for which ||x -- a|| is the kth smallest among the Euclidean distances between x and the points in A. If A is a lattice, the kth Brillouin zones of the points in A are translates of each other, and together they tile space. Depending on the value of k, they express medium- or long-range order in the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone under perturbations, a linear upper bound on the number of chambers in a zone for lattices in R², and the convergence of the maximum volume of a chamber to zero for the integer lattice. [ABSTRACT FROM AUTHOR]