## Abstract

We prove that if a closed planar set S is not a countable union of convex subsets, then exactly one of the following holds: (a) There is a perfect subset P⊆S such that for every pair of distinct points x, yεP, the convex closure of x, y is not contained in S. (b) (a) does not hold and there is a perfect subset P⊆S such that for every pair of points x, yεP the convex closure of {x, y} is contained in S, but for every triple of distinct points x, y, zεP the convex closure of {x, y, z} is not contained in S. We show that an analogous theorem is impossible for dimension greater than 2. We give an example of a compact planar set with countable degree of visual independence which is not a countable union of convex subsets, and give a combinatorial criterion for a closed set in R^{ d} not to be a countable union of convex sets. We also prove a conjecture of G. Kalai, namely, that a closed planar set with the property that each of its visually independent subsets has at most one accumulation point, is a countable union of convex sets. We also give examples of sets which possess a (small) finite degree of visual independence which are not a countable union of convex subsets.

Original language | English |
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Pages (from-to) | 313-342 |

Number of pages | 30 |

Journal | Israel Journal of Mathematics |

Volume | 70 |

Issue number | 3 |

DOIs | |

State | Published - 1 Oct 1990 |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics (all)