An estimate of the centroid Banach-Mazur distance between planar convex bodies

TL;DR

This paper establishes an upper bound of 69/17 for the centroid Banach-Mazur distance between any two convex bodies in the plane, considering the additional centroid coincidence constraint.

Contribution

It introduces and analyzes the centroid-constrained Banach-Mazur distance, providing a specific universal bound in two-dimensional space.

Findings

Bound of 69/17 for planar convex bodies with coinciding centroids

Extension of Banach-Mazur distance concept with centroid condition

Theoretical proof of the bound in 2D case

Abstract

We consider the variant of the Banach-Mazur distance $\delta_{BM}^{\rm cen} (C, D)$ of two convex bodies $C, D$ of $E^d$ with the additional requirement that the centroids of them coincide. We prove that $\delta_{BM}^{\rm cen} (C, D) \leq \frac{69}{17}$ for every $C, D$ of $E^2$.