Cantor-Bendixson degrees and convexity in ℝ2

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4 Scopus citations


We present an ordinal rank, δ3, which refines the standard classification of non-convexity among closed planar sets. The class of closed planar sets falls into a hierarchy of order type ω1 + 1 when ordered by δ-rank. The rank δ3 (S) of a set S is defined by means of topological complexity of 3-cliques in the set. A 3-clique in a set S is a subset of 5 all of whose unordered 3-tuples fail to have their convex hull in S. Similarly, δn(S) is defined for all n > 1. The classification cannot be done using δ2, which considers only 2-cliques (known in the literature also as "visually independent subsets"), and in dimension 3 or higher the analogous classification is not valid.

Original languageEnglish
Pages (from-to)85-91
Number of pages7
JournalIsrael Journal of Mathematics
StatePublished - 1 Jan 2001

ASJC Scopus subject areas

  • Mathematics (all)


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