On Convex Curves Which Have Many Inscribed Triangles of Maximum Area

Let K be a convex figure in the plane such that every point x is an element of partial derivative K serves as a vertex of an inscribed triangle with maximum area. In this note, we prove a conjecture due to Genin and Tabachnikov that says 3 root 3/4 pi <= vertical bar T vertical bar/vertical bar K vertical bar <= 1/2, where T is a triangle with maximum area inscribed in K. Moreover, we prove that the bounds in the left side and the right side of the inequality are obtained only for ellipses and parallelograms, respectively.