Extremal Banach-Mazur distance between a symmetric convex body and an arbitrary convex body on the plane

TL;DR

This paper characterizes convex bodies in the plane with a Banach-Mazur distance of 2 from a symmetric convex body, showing such bodies must be triangles, thus providing an extremal geometric property.

Contribution

It establishes a precise geometric characterization of convex bodies at the extremal Banach-Mazur distance from symmetric bodies in the plane.

Findings

Convex bodies at distance 2 from symmetric bodies are triangles.

The result is sharp and characterizes extremal cases.

Provides insight into the geometry of convex bodies and Banach-Mazur distances.

Abstract

We prove that if $K, L \subset \mathbb{R}^2$ are convex bodies such that $L$ is symmetric and the Banach-Mazur distance between $K$ and $L$ is equal to $2$, then $K$ is a triangle.