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

arXiv: https://arxiv.org/abs/2501.02281 · HTML: https://mlsanity.com/paper/2501.02281/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jan 2025.

## Built on by (most cited first, 1 of 1)

- [Numerical exploration of the range of shape functionals using neural networks](https://mlsanity.com/paper/2602.14881/) (2026)

## Abstract (arXiv)

The object of the paper is to find complete systems of inequalities relating the perimeter P, the area |·| 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,|·|) 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≥ 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.