Circumscribed Polygons of Small Area

Given any plane strictly convex region K and any positive integer n≥3, there exists an inscribed 2n-gon Q _2n and a circumscribed n-gon P _n such that The inequality is the best possible, as can be easily seen by letting K be an ellipse. As a corollary, it follows that for any convex region K and any n≥3, there exists a circumscribed n-gon P _n such that This improves the existing bounds for 5≤n≤11.