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Separation of two convex sets in convexity structures

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Abstract

A convexity structure satisfies the separation propertyS 4 if any two disjoint convex sets extend to complementary half-spaces. This property is investigated for alignment spaces,n-ary convexities, and graphs. In particular, it is proven that

  1. a)

    ann-ary convexity isS 4 iff every pair of disjoint polytopes with at mostn vertices can be separated by complementary half spaces, and

  2. b)

    an interval convexity isS 4 iff it satisfies the analogue of the Pasch axiom of plane geometry.

A characterization of bipartite and weakly modular spaces withS 4 convexity is given in terms of forbidden subgraphs.

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Authors and Affiliations

  1. Department of Mathematics and Cybernetics, Moldova State University, A.Mateevici str. 60, 277009, Chişinău, Moldova

    Victor Chepoi

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  1. Victor Chepoi
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Dedicated to Professor N.K. Stephanidis, on the occasion of his 65 birthday

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Chepoi, V. Separation of two convex sets in convexity structures. J Geom 50, 30–51 (1994). https://doi.org/10.1007/BF01222661

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  • DOI: https://doi.org/10.1007/BF01222661

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