Bounding the diameter-width ratio using containment inequalities of means of convex bodies

TL;DR

This paper characterizes the relationship between diameter and width ratios of planar convex sets using containment inequalities involving Minkowski differences and asymmetry measures.

Contribution

It provides a complete description of the possible diameter-width ratios for planar pseudo-complete sets based on Minkowski asymmetry and containment inequalities.

Findings

Derived bounds for the diameter-width ratio in terms of Minkowski asymmetry.

Established the region of possible values of the containment factor τ(K).

Connected geometric inequalities with Minkowski asymmetry in the planar case.

Abstract

We completely describe the region of possible values of the diameter-width ratio for planar pseudo-complete sets in dependence of the Minkowski asymmetry. In order to do this, we focus on the containment inequalities of $K \cap (-K)$ and $\frac{K-K}{2}$ for a Minkowski centered convex compact set $K$, i.e. we define $\tau(K)$ to be the smallest possible factor to cover $K \cap (-K)$ by a rescalation of $\frac{K-K}{2}$ and give the region of the possible values of $\tau(K)$ in the planar case in dependence of the Minkowski asymmetry of $K$.