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

Ilias Ftouhi

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arXiv · 2501.02281

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

Ilias Ftouhi

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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.

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BibTeX RIS

Ilias Ftouhi. 2025-01-04. Complete systems of inequalities relating the perimeter, the area and the Cheeger constant of planar domains. https://arxiv.org/abs/2501.02281

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