Minimal Convex Decompositions

Let P be a set of n points on the plane in general position. We say that a set Γ of convex polygons with vertices in P is a convex decomposition of P if: Union of all elements in Γ is the convex hull of P, every element in Γ is empty, and for any two different elements of Γ their interiors are disjoint. A minimal convex decomposition of P is a convex decomposition Γ′ such that for any two adjacent elements in Γ′ its union is a non convex polygon. It is known that P always has a minimal convex decomposition with at most 3n 2 elements. Here we prove that P always has a minimal convex decomposition with at most 10n 7 elements.