In this paper we study the problem of partitioning point sets in the p
lane so that each equivalence class is a convex polygon with some cond
itions on the intersection properties of such sets. Let P be a set of
n points in the plane. We define f(P) to be the minimum number of sets
in a partition into disjoint convex polygons of P and F(n) as the max
imum of f(P), over all sets P of n points. We establish lower and uppe
r bounds for F(n). We also estimate the maximum of the minimum number
of sets in a partition into empty convex polygons, over all sets of n
points. Finally, we consider the maximum of the minimum number of conv
ex polygons which cover the n points set P, over all sets P of n point
s.