Non-Convexity

TL;DR

This paper extends Valentine’s 1957 result by showing that any planar set where every three points have at least two seeing each other can be decomposed into at most six convex sets, removing the closedness condition.

Contribution

It generalizes Valentine’s theorem by removing the closedness condition and establishing a tight upper bound of six convex sets for such planar sets.

Findings

Any such set can be decomposed into at most six convex sets.

The bound of six convex sets is optimal.

The result generalizes previous work by relaxing topological constraints.

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.