On the Regions Formed by the Diagonals of a Convex Polygon

C P Anil Kumar

Published 2019-04-05Version 1

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. Here in this article we associate to every region a unique $n$-cycle in the symmetric group $S_n$ of a certain type (defined as two 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 main Theorem $\Omega$ 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.