In 2001 K. Hosono and M. Urabe
considered the problem of
the number of disjoint empty convex k-gons in a planar point
set for a fixed k. They mainly studied the case k=4 and proved F4(9)=2, 
. In this paper, we remove the restriction of disjointness and
consider a
related problem: how many empty convex k-gons can be constructed in a planar
point set for a fixed k? We mainly study case of k=4 as well and
obtain
some meaningful results.
Let P be a set of n points in the plane with no three
collinear, that is, P is in general position. We call a
partition of P an {\it empty convex partition} if P is partitioned
into subsets S1, S2, ..., St; 
such
that each CH(Si) is an |Si|-gon for every i, and CH(Si)
is empty, where CH denotes the convex hull. Let k be a positive
integer and 
be the number of empty convex
k-gons in an empty convex partition 
of P. We
denote
gk(
P)=:
max {
:
is an empty convex partition of
P.}
Gk(
n)=:
min{{
gk(
P): |
P|=
n}
In this paper, we obtain the following results :
G4(5)=1, G4(9)=2, G4(13)=3, G4(17)=4
By using these results we show that for a set of
38 points we can construct 9 empty convex quadrilaterals and
so we obtain 
In the proof, we make use of the following
lemma proved by K.Hosono and M.Urabe:
For any set of 2m+4 points in the plane, no three collinear, we
can divide the plane
into three disjoint convex regions such that one contains a
convex quadrilateral and the others contain m points each,
where m is a positive integer.
Moreover, for 
, we get the following better bound:
.
