On the maximum area of inscribed polygons

Abstract

Given a convex $n$-gon $P$ and a positive integer $m$ such that $3\le m\le n-1$, let $Q$ denote the largest area convex $m$-gon contained in $P$. We are interested in the minimum value of $\Delta(Q)/\Delta(P)$, the ratio of the areas of these two polygons. More precisely, given positive integers $n$ and $m$, with $3 \le m \le n-1$, define \begin{equation*} f_n(m)=\min_{P\in \mathcal {P}_n} \max_{Q \subset P,|Q|=m} \frac{\Delta(Q)}{\Delta(P)} \end{equation*} where the maximum is taken over all $m$-gons contained in $P$, and the minimum is taken over $\mathcal{P}_n$, the entire class of convex $n$-gons. The values of $f_4(3)$, $f_5(4)$ and $f_6(3)$ are known. In this paper we compute the values of $f_5(3)$, $f_6(5)$ and $f_6(4)$. In addition, we prove that for all $n\ge 6$ we have \begin{equation*} \frac{4}{n}\cdot\sin^2\left(\frac{\pi}{n}\right)\le 1-f_n(n-1)\le \min\left(\frac{1}{n}, \frac{4}{n}\cdot\sin^2\left(\frac{2\pi}{n}\right)\right). \end{equation*} These bounds can be used to improve the known estimates for $f_n(m)$.