Let P be a set of finite points
in the plane, no three collinear. A convex polygon determined by a
subset of P is called an empty polygon if no points of P lie
in the interior of its convex hull. More generally, we call a
closed region D empty, denoted by D 
,if no points of P lie in the interior of D.
A partition of P is called a convex partition if P is partitioned
by k subsets S1 , S2 ,..., Sk, such that each
CH (Si ) is an |Si |-gon, where CH( Si ) denotes the convex hull
of Si . If CH( Si ) 
CH( Sj ) = for any pair of indices ( i 
j ), then this partition is called a
disjoint partition of P; if CH( Si ) for each i, we
call this partition an empty partition of P.
Let k be a positive integer and 
be the number
of convex k-gons in a partition 
of P. Let 
be
the number of convex polygons in a partition of P. We
denote
fk(P) = : max{
:
is a disjiont partition of
P}
Fk(n) = :min{
fk(P):|P|=n}
f(P) = : min{
:
is a disjiont partition of
P}
F(n) = :max
{f(P):|P|=n}
g(P) = : min{
:
is an empty partition of
P}
G(n) = :max{
g(P):|P|=n}
Hosono and Urabe studied these functions,
and we improve the upper bound of G(n).
Moreover, we define
gk(P) = : max{
:
is an empty partition of
P}
Gk(n) = :min{
gk(P):|P|=n}
and obtain the following results:
