Sharp inequalities involving the Cheeger constant of planar convex sets

TL;DR

This paper establishes sharp bounds for the Cheeger constant of planar convex sets using various geometric measures, providing complete solutions for certain inequality diagrams and proposing new conjectures.

Contribution

It introduces new sharp inequalities involving the Cheeger constant and geometric quantities, and fully characterizes the Blaschke-Santaló diagrams for key triplets.

Findings

Complete solutions for the diagrams involving (P,h,r), (d,h,r), and (R,h,r).

Partial descriptions of boundaries for diagrams involving (ω,h,·).

New conjectures based on numerical simulations.

Abstract

We are interested in finding sharp bounds for the Cheeger constant $h$ via different geometrical quantities, namely the area $|\cdot|$, the perimeter $P$, the inradius $r$, the circumradius $R$, the minimal width $\omega$ and the diameter $d$. We provide new sharp inequalities between these quantities for planar convex bodies and enounce new conjectures based on numerical simulations. In particular, we completely solve the Blaschke-Santal\'o diagrams describing all the possible inequalities involving the triplets $(P,h,r)$, $(d,h,r)$ and $(R,h,r)$ and describe some parts of the boundaries of the diagrams of the triplets $(\omega,h,d)$, $(\omega,h,R)$, $(\omega,h,P)$, $(\omega,h,|\cdot|)$, $(R,h,d)$ and $(\omega,h,r)$.