The Centroid Banach-Mazur Distance between the Parallelogram and the Triangle

Let

and

D D

be convex bodies in the Euclidean space

E^d E d

. We define the centroid Banach-Mazur distance

\delta_{BM}^{\rm cen} (C, D) δ B M c e n ( C , D )

similarly to the classic Banach-Mazur distance

\delta_{BM} (C, D) δ B M ( C , D )

, but with the extra requirement that the centroids of

C C

and an affine image of

D D

coincide. We prove that for the parallelogram

P P

and the triangle

T T

in

E^2 E 2

we have

\delta_{BM}^{\rm cen} (P, T) = \frac{5}{2} δ B M c e n ( P , T ) = 5 2

.