Position of the centroid of a planar convex body

Abstract

It is well known that any planar convex body $A$ permits to inscribe an affine-regular hexagon $H_A$. We prove that the centroid of $A$ belongs to the homothetic image of $H_A$ with ratio $\frac{4}{21}$ and the center in the center of $H_A$. This ratio cannot be enlarged.