Blaschke--Santal{ó} diagram for the volume, the diameter, and the Cheeger constant

Abstract

We study the Blaschke--Santal{ó} diagram associated with the volume, the diameter, and the Cheeger constant. In the class of bounded open subsets of $\mathbb{R}^m$, $m\ge 2$, we give a complete description of the diagram. In the class of convex bodies of $\mathbb R^m$, we prove that the diagram is closed and simply connected as it is given by the region between two continuous functions for which qualitative properties such as monotonicity, local behavior, and growth estimates are studied.