On Polygons Enclosing Point Sets II

Let R and B be disjoint point sets such that R ∪ B is in general position. We say that B encloses by R if there is a simple polygon P with vertex set B such that all the elements in R belong to the interior of P . In this paper we prove that if the vertices of the convex hull of R ∪ B belong to B, and |R| ≤ |Conv(B)| − 1 then B encloses R. The bound is tight. This improves on results of a previous paper in which it was proved that if |R| ≤ 56|Conv (B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices p1, . . . , pn and S a set of m points contained in the interior of P , m ≤ n− 1. Then there is a convex decomposition {P1, . . . , Pn} of P such that all points from S lie on the boundaries of P1, . . . , Pn, and each Pi contains a whole edge of P on its boundary.