On the smallest area $(n-1)$-gon containing a convex n-gon

Abstract

Approximation of convex disks by inscribed and circumscribed polygons is a classical geometric problem whose study is motivated by various applications in robotics and computer aided design. We consider the following optimization problem: given integers $3\le n\le m-1$, find the value or an estimate of

$$\begin{aligned} r(n,m)=\max _{P\in {\mathcal {P}}_m}\,\, \min _{Q\in {\mathcal {P}}_n,\,Q \supseteq P} \frac{|Q|}{|P|} \end{aligned}$$

where P varies in the set ${\mathcal {P}}_m$ of all convex m-gons, and, for a fixed m-gon P, the minimum is taken over all n-gons Q containing P; here $|\cdot |$ denotes area. It is easy to prove that $r(3,4)=2$, and from a result of Gronchi and Longinetti it follows that $r(n-1, n)= 1+\frac{1}{n}\tan \left( \pi /{n}\right) \tan \left( {2\pi }/{n}\right)$ for all $n\ge 6$. In this paper we show that every unit area convex pentagon is contained in a convex quadrilateral of area no greater than $3/\sqrt{5}$ thus determining the value of r(4, 5). In all cases, the equality is reached only for affine regular polygons.