(Hebrew University of Jerusalem)
"A planar 3-convex set is indeed a union of six convex sets"
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.