Banach-Mazur distances between parallelograms and other affinely regular even-gons

TL;DR

This paper determines the Banach-Mazur distance between parallelograms and affine-regular even-gons, establishing the maximum distance as 1.5 for centrally-symmetric planar convex bodies, and provides new proofs for these geometric bounds.

Contribution

It offers the first published proof of the Banach-Mazur distance between parallelograms and affine-regular hexagons, and extends the analysis to other affine-regular even-gons.

Findings

Banach-Mazur distance between parallelogram and affine-regular hexagon is 1.5

Diameter of centrally-symmetric planar convex bodies is 1.5

New proof for the maximum Banach-Mazur distance in this context

Abstract

We show that the Banach-Mazur distance between the parallelogram and the affine-regular hexagon is $\frac{3}{2}$ and we conclude that the diameter of the family of centrally-symmetric planar convex bodies is just $\frac{3}{2}$. 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 $\frac{3}{2}$. Analogously, we deal with the Banach-Mazur distances between the parallelogram and the remaining affine-regular even-gons.