Area properties associated with a convex plane curve

Archimedes knew that for a point P on a parabola X and a chord AB of X parallel to the tangent of X at P, the area of the region bounded by the parabola X and chord AB is four thirds of the area of the triangle Delta ABP. Recently, the first two authors have proved that this fact is the characteristic property of parabolas. In this paper, we study strictly locally convex curves in the plane R-2. As a result, generalizing the above mentioned characterization theorem for parabolas, we present two conditions, which are necessary and sufficient, for a strictly locally convex curve in the plane to be an open arc of a parabola.