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

Abstract

The object of the paper is to find complete systems of inequalities relating the perimeter [Formula: see text], the area [Formula: see text] and the Cheeger constant [Formula: see text] of planar sets. To do so, we study the so-called Blaschke–Santaló diagram of the triplet [Formula: see text] for different classes of domains: simply connected sets, convex sets and convex polygons with at most [Formula: see text] sides. We completely determine the diagram in the latter cases except for the class of convex [Formula: see text]-gons when [Formula: see text] 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.