  For a positive integer $n\geq 3$, the sides and diagonals of a convex $n$-gon
divide the interior of the convex $n$-gon into finitely (polynomial in $n$)
many regions bounded by them. In this article, we associate to every region a
unique $n$-cycle in the symmetric group $S_n$ of a certain type (defined as
$2$-standard consecutive cycle) by studying point arrangements in the plane.
Then we find that there are more (exponential in $n$) number of such cycles
leading to the conclusion that not every region labelled by a cycle appears in
every convex $n$-gon. In fact most of them do not occur in any given single
convex $n$-gon. Later in the main theorem of this article we characterize
combinatorially those cycles (defined as definite cycles) whose corresponding
regions occur in every convex $n$-gon and those cycles (defined as indefinite
cycles) whose corresponding regions do not occur in every convex $n$-gon. As a
consequence we characterize those one point extensions of a uniform rank $3$
convex oriented matroid for which the one point extension is reducible by the,
one point, when it lies inside the convex hull.
