## Abstract

We present an ordinal rank, δ^{3}, 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 ω_{1} + 1 when ordered by δ-rank. The rank δ^{3} (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, δ^{n}(S) is defined for all n > 1. The classification cannot be done using δ^{2}, 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 language | English |
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Pages (from-to) | 85-91 |

Number of pages | 7 |

Journal | Israel Journal of Mathematics |

Volume | 121 |

DOIs | |

State | Published - 1 Jan 2001 |

## ASJC Scopus subject areas

- Mathematics (all)

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