Sharp inequalities involving the Cheeger constant of planar convex sets

¹ Friedrich-Alexander-Universität Erlangen-Nürnberg, Department of Mathematics, Chair of Applied Analysis (Alexander von Humboldt Professorship), Cauerstr. 11, 91058 Erlangen, Germany
² Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Universita degli Studi di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, via Cintia, 80126 Napoli, Italy
³ Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Universita degli Studi di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, via Cintia, 80126 Napoli, Italy

^* Corresponding author: albalia.masiello@unina.it

Received: 22 June 2023
Accepted: 27 February 2024

Abstract

We are interested in finding sharp bounds for the Cheeger constant h via different geometrical quantities, namely the area |•|, the perimeter P, the inradius r, the circumradius R, the minimal width w 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-Santaló 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 (w, h, d), (w, h, R), (w, h, P), (w, h, |•|), (R, h, d) and (w, h, r).