The combinatorial encoding of disjoint convex sets in the plane

Abstract

We introduce a new combinatorial object, the double-permutation sequence, and use it to encode a family of mutually disjoint compact convex sets in the plane in a way that captures many of its combinatorial properties. We use this encoding to give a new proof of the Edelsbrunner-Sharir theorem that a collection of n compact convex sets in the plane cannot be met by straight lines in more than 2n-2 combinatorially distinct ways. The encoding generalizes the authors’ encoding of point configurations by “allowable sequences” of permutations. Since it applies as well to a collection of compact connected sets with a specified pseudoline arrangement \( \mathcal{A} \) of separators and double tangents, the result extends the Edelsbrunner-Sharir theorem to the case of geometric permutations induced by pseudoline transversals compatible with \( \mathcal{A} \).

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Correspondence to Jacob E. Goodman.

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Supported in part by NSA grant MDA904-03-I-0087 and PSC-CUNY grant 65440-0034.

Supported in part by NSF grant CCR-9732101.

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Goodman, J.E., Pollack, R. The combinatorial encoding of disjoint convex sets in the plane. Combinatorica 28, 69–81 (2008). https://doi.org/10.1007/s00493-008-2239-7

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Mathematics Subject Classification (2000)

  • 52A35
  • 52C30