Sharp inequalities involving the Cheeger constant of planar convex sets

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 $(\omega,h,r)$.