On Banach-Mazur distance between planar convex bodies

TL;DR

This paper improves bounds on the Banach-Mazur distance between planar convex bodies, providing tighter estimates for the diameter and radius of their family, advancing understanding in convex geometry.

Contribution

It offers new upper bounds for the diameter and radius of planar convex bodies under the Banach-Mazur distance, refining previous estimates.

Findings

Diameter does not exceed approximately 2.614.

Radius does not exceed approximately 1.671.

Improves previous bounds from 3 to tighter estimates.

Abstract

Upper estimates of the diameter and the radius of the family of all planar convex bodies with respect to the Banach-Mazur distance are obtained. Namely, it is shown that the diameter does not exceed $\tfrac{19-\sqrt{73}}4\approx 2.614$, which improves the previously known bound of $3$, and that the radius does not exceed $\frac{117}{70}\approx 1.671$.