On the Maximal Distance Between the Centers of Mass of a Planar Convex Body and Its Boundary

Abstract

We prove that the length of the projection of the vector joining the centers of mass of a convex body on the plane and of its boundary to an arbitrary direction does not exceed \(\frac{1}{6}\) of the body width in this direction. It follows that the distance between these centers of mass does not exceed \(\frac{1}{6}\) of the diameter of the body and \(\frac{1}{12}\) of its boundary length. None of those constants can be improved.