On the maximal distance between the centers of mass of a planar convex body and its boundary

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.