Given a pair of distinct points p and q in a metric space with distanced, the mediatrix is the set of points x such that d(x, p) = d(x, q). In this paper, we examine the topological structure of mediatrices in connected, compact, closed 2-manifolds whose distance function is inherited from a Riemannian metric. We determine that such mediatrices are, up to homeomorphism, finite, closed simplicial 1-complexes with an even number of incipient edges emanating from each vertex. Using this and results from [J.J.P. Veerman, J. Bernhard, Minimally separating sets, mediatrices and Brillouin spaces, Topology Appl., in press], we give the classification up to homeomorphism of mediatrices on genus 1 tori (and on projective planes) and outline a method which may possibly be used to classify mediatrices on higher-genus surfaces.
The Topology Of Surface MediatricesTopology And Its Applications
DepartmentMathematics and Computer Science
Citation InformationBernhard, James, and J. J. P. Veerman. 2007. "The topology of surface mediatrices." Topology And Its Applications 154(1): 54-68.