A Jordan–Brouwer Separation Theorem for Polyhedral Pseudomanifolds

  • Authors:

  • Micha A. Perles;Horst Martini;Yaakov S. Kupitz

  • Affiliations:

  • The Hebrew University of Jerusalem, Institute of Mathematics, Jerusalem, Israel;University of Technology, Faculty of Mathematics, 09107, Chemnitz, Germany;The Hebrew University of Jerusalem, Institute of Mathematics, Jerusalem, Israel

  • Venue:
  • Discrete & Computational Geometry - Special Issue Dedicated to the Memory of Victor Klee
  • Year:
  • 2009

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Abstract

The Jordan Curve Theorem referring to a simple closed curve in the plane has a particularly simple proof in the case that the curve is polygonal, called the “raindrop proof”. We generalize the notion of a simple closed polygon to that of a polyhedral (d−1)-pseudomanifold (d≥2) and prove a Jordan–Brouwer Separation Theorem for such a manifold embedded in ℝd . As a by-product, we get bounds on the polygonal diameter of the interior and exterior of such a manifold which are almost tight. This puts the result within the frame of computational geometry.