The centroid Banach-Mazur distance between the parallelogram and the triangle

Abstract

Let $C$ and $D$ be convex bodies in the Euclidean space $E^d$. We define the centroid Banach-Mazur distance ${\delta }_{BM}^{\mathrm{cen}}(C,D)$ similarly to the classic Banach-Mazur distance ${\delta }_{BM}(C,D)$, but with the extra requirement that the centroids of $C$ and an affine image of $D$ coincide. We prove that for the parallelogram $P$ and the triangle $T$ in $E^2$ we have ${\delta }_{BM}^{\mathrm{cen}}(P,T)=\frac{5}{2}$.

8 pages, 6 figures