Complete systems of inequalities relating the perimeter, the area and the Cheeger constant of planar domains

TL;DR

This paper characterizes the relationships between perimeter, area, and Cheeger constant of planar domains through inequalities, providing complete diagrams for various classes and applications of these geometric bounds.

Contribution

It offers a comprehensive analysis of inequalities relating perimeter, area, and Cheeger constant for different classes of planar sets, including explicit formulas and numerical methods.

Findings

Complete characterization of the inequality diagrams for convex polygons.

Explicit formula for the lower boundary of the diagram.

Numerical method for the upper boundary of the diagram.

Abstract

The object of the paper is to find complete systems of inequalities relating the perimeter $P$, the area $|\cdot|$ and the Cheeger constant $h$ of planar sets. To do so, we study the so called Blaschke--Santal\'o diagram of the triplet $(P,h,|\cdot|)$ for different classes of domains: simply connected sets, convex sets and convex polygons with at most $N$ sides. We completely determine the diagram in the latter cases except for the class of convex $N$-gons when $N\ge 5$ is odd: therein, we show that the boundary of the diagram is given by the graphs of two continuous and strictly increasing functions. An explicit formula for the lower one and a numerical method to obtain the upper one is provided. At last, some applications of the results are presented.