Planar Point Sets With Large Minimum Convex Decompositions

We show the existence of sets with n points (n > 4) for which every convex decomposition contains more than f§« — § polygons, which refutes the conjecture that for every set of n points there is a convex decomposition with at most n + C polygons. For sets having exactly three extreme points we show that more than n + s/2(n 3) 4 polygons may be necessary to form a convex decomposition.