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

Marek Lassak

Published 2020-08-04Version 1

First we explain the positions of the parallelogram with respect to a given centrally symmetric planar convex body $C$ which realize the Banach-Mazur distance to $C$. Next we prove that the Banach-Mazur distance from the parallelogram to the affinely regular hexagon is $\frac{3}{2}$, showing also all the optimal positions of the parallelogram with respect to the hexagon. Analogously, we deal with the distances to the remaining affinely regular even-gons. Namely, we find the distances to the affinely regular $8j$-gons and $(8j+4)$-gons. Moreover, we estimate and conjecture the distances to the affinely regular $(8j+2)$-gons and $(8j+6)$-gons.