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

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.