An estimate of the centroid Banach-Mazur distance between planar convex bodies

Abstract: We consider the variant of the Banach-Mazur distance $\delta_{BM}^{\rm cen} (C, D)$ of two convex bodies $C, D$ of $E^d$ with the additional requirement that the centroids of them coincide. We prove that $\delta_{BM}^{\rm cen} (C, D) \leq \frac{69}{17}$ for every $C, D$ of $E^2$.