Banach–Mazur Distance from the Parallelogram to the Affine-Regular Hexagon and Other Affine-Regular Even-Gons

Abstract

AbstractWe show that the Banach–Mazur distance between the parallelogram and the affine-regular hexagon is $$\frac{3}{2}$$ 3 2 and we conclude that the diameter of the family of centrally-symmetric planar convex bodies is just $$\frac{3}{2}$$ 3 2 . A proof of this fact does not seem to be published earlier. Asplund announced this without a proof in his paper proving that the Banach–Mazur distance of any planar centrally-symmetric bodies is at most $$\frac{3}{2}$$ 3 2 . Analogously, we deal with the Banach–Mazur distances between the parallelogram and the remaining affine-regular even-gons.