The affine representation theorem for abstract convex geometries

  • Authors:

  • Kenji Kashiwabara;Masataka Nakamura;Yoshio Okamoto

  • Affiliations:

  • Department of Systems Science, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1, Komaba, Meguro, Tokyo, 153-8902, Japan;Department of Systems Science, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1, Komaba, Meguro, Tokyo, 153-8902, Japan;Institute of Theoretical Computer Science, Department of Computer Science, ETH Zurich, CH-8092 Zurich, Switzerland

  • Venue:
  • Computational Geometry: Theory and Applications - Special issue on the 19th European workshop on computational geometry - EuroCG 03
  • Year:
  • 2005

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Abstract

A convex geometry is a combinatorial abstract model introduced by Edelman and Jamison which captures a combinatorial essence of ''convexity'' shared by some objects including finite point sets, partially ordered sets, trees, rooted graphs. In this paper, we introduce a generalized convex shelling, and show that every convex geometry can be represented as a generalized convex shelling. This is ''the representation theorem for convex geometries'' analogous to ''the representation theorem for oriented matroids'' by Folkman and Lawrence. An important feature is that our representation theorem is affine-geometric while that for oriented matroids is topological. Thus our representation theorem indicates the intrinsic simplicity of convex geometries, and opens a new research direction in the theory of convex geometries.