Non-Convexity
Combinatorics
2009-01-28 v1
Abstract
Suppose S is a planar set. Two points a,b in S 'see each other' via S if [a,b] is included in S . F. Valentine proved in 1957 that if S is closed, and if for every three points of S, at least two see each other via S, then S is a union of three convex sets. The pentagonal star shows that the number three is best possible. We discard the condition that S is closed and show that S is a union of (at most) six convex sets. The number six is best possible.
Cite
@article{arxiv.0901.4139,
title = {Non-Convexity},
author = {Noa Nitzan},
journal= {arXiv preprint arXiv:0901.4139},
year = {2009}
}