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

Abstract

We are interested in finding complete systems of inequalities between the perimeter P , the area | • | and the Cheeger constant h of planar sets. To do so, we study the so called Blaschke-Santaló diagram of the triplet (P, | • |, h) for different classes of domains: simply connected sets, convex sets and convex polygons with at most N sides. We are able to completely determine the diagram in the latter cases except for the class of convex N -gons when N ≥ 5 is odd: therein, we show that the external boundary of the diagram is given by the curves of two continuous and strictly increasing functions, we give an explicit formula of the lower one and provide a numerical method to obtain the upper one. We finally give some applications of the results and methods developed in the present paper.

Isoperimetric Sets and p-Cheeger Sets are in Bijection

“…We refer to [21] for convex sets, to [22,25] for strips, and to [26,28] for the most general statement. In dimension 2, additional properties have been proved when enjoys a rotational symmetry [7], and we also mention that a complete characterization of the Blaschke-Santaló diagram for the triplet Cheeger constant, perimeter, and area of has been recently obtained in [16], and for more general triplets in [18]. Finally, some stability results in the planar case are available in [11].…”