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The Topology Of Surface Mediatrices
Topology And Its Applications
  • James Bernhard, University of Puget Sound
  • J. J. P. Veerman
Document Type
Publication Date
Mathematics and Computer Science

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.

Citation Information
Bernhard, James, and J. J. P. Veerman. 2007. "The topology of surface mediatrices." Topology And Its Applications 154(1): 54-68.