Minimally Separating Sets, Mediatrices, and Brillouin Spaces

Brillouin zones and their boundaries were studied in [16] because they play an important role in focal decomposition as first defined by Peixoto in [9] and in physics ([1] and [3]). In so-called Brillouin spaces, the boundaries of the Brillouin zones have certain regularity properties which imply that they consist of pieces of mediatrices (or equidistant sets). The purpose of this note is two-fold. First, we give some simple conditions on a metric space which are sufficient for it to be a Brillouin space. These conditions show, for example, that all compact, connected Riemannian manifolds with their usual distance functions are Brillouin spaces. Second, we exhibit a restriction on the Z2-homology of mediatrices in such manifolds in terms of the Z2-homology of the manifolds themselves, based on the fact that they are Brillouin spaces. (This will used to obtain a classification up to homeomorphim of surface mediatrices in [15]). This note begins with some preliminaries in Section 1, where we define the relevant concepts and provide some background and motivation for the questions being considered here. In Section 2, we list a simple set of conditions on a metric space and prove that these conditions suffice for it to be a Brillouin space. In Section 3, we give some examples and mention a result for use in a later paper. We then investigate in Section 4 the homological restrictions placed on a mediatrix by the topology of the surrounding Brillouin space. The paper then closes with some concluding remarks in Section 5 and some acknowledgements.