%0 Journal Article
%T Non-Convexity
%A Noa Nitzan
%J Mathematics
%D 2009
%I arXiv
%X 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.
%U http://arxiv.org/abs/0901.4139v1