On the maximum area of inscribed polygons

Abstract

Given a convex n-gon P and a positive integer m such that 3 ≤ m ≤ n − 1, let Q denote the largest area convex m-gon contained in P . We are interested in the minimum value of ∆(Q)/∆(P ), the ratio of the areas of these two polygons. More precisely, given positive integers n and m, with 3 ≤ m ≤ n− 1, define fn(m) = min P∈Pn max Q⊂P,|Q|=m ∆(Q) ∆(P ) where the maximum is taken over all m-gons contained in P , and the minimum is taken over Pn, the entire class of convex n-gons. The values of f4(3), f5(4) and f6(3) are known. In this paper we compute the values of f5(3), f6(5) and f6(4). In addition, we prove that for all n ≥ 6 we have 4 n · sin (π n ) ≤ 1− fn(n− 1) ≤ min (