MEMO 2015 I2

Let $n\ge 3$ be an integer. An inner diagonal of a simple $n$ -gon is a diagonal that is contained in the $n$ -gon. Denote by $D(P)$ the number of all inner diagonals of a simple $n$ -gon $P$ and by $D(n)$ the least possible value of $D(Q)$ , where $Q$ is a simple $n$ -gon. Prove that no two inner diagonals of $P$ intersect (except possibly at a common endpoint) if and only if $D(P)=D(n)$ .

Remark: A simple $n$ -gon is a non-self-intersecting polygon with $n$ vertices. A polygon is not necessarily convex.