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

Ilias Ftouhi

TL;DR

The paper analyzes complete systems of inequalities among the perimeter $P$, area $|\Omega|$, and Cheeger constant $h(\Omega)$ for planar sets via Blaschke–Santaló diagrams. It provides exact diagram descriptions for simply connected and convex sets, and a detailed boundary characterization for convex polygons with $N$ sides, including explicit upper bounds when $N$ is even and a piecewise, numerically informed boundary for odd $N\ge5$ with asymptotic slope $g_N(x)\sim x$. Numerical simulations for pentagons validate the boundary structures and reveal the behavior of non–Cheeger-regular extremals, while applications yield a sharper lower bound for polygon Cheeger constants and a quantitative stability result in terms of the Hausdorff distance. Overall, the work unifies geometric inequalities for $P$, $|\Omega|$, and $h(\Omega)$ across several planar classes and provides both exact and computational tools to describe the feasible region of these functionals. The results have potential impact on shape optimization, geometric analysis, and the study of isoperimetric-type inequalities in the plane.

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--Santaló 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.

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

TL;DR

The paper analyzes complete systems of inequalities among the perimeter , area , and Cheeger constant for planar sets via Blaschke–Santaló diagrams. It provides exact diagram descriptions for simply connected and convex sets, and a detailed boundary characterization for convex polygons with sides, including explicit upper bounds when is even and a piecewise, numerically informed boundary for odd with asymptotic slope . Numerical simulations for pentagons validate the boundary structures and reveal the behavior of non–Cheeger-regular extremals, while applications yield a sharper lower bound for polygon Cheeger constants and a quantitative stability result in terms of the Hausdorff distance. Overall, the work unifies geometric inequalities for , , and across several planar classes and provides both exact and computational tools to describe the feasible region of these functionals. The results have potential impact on shape optimization, geometric analysis, and the study of isoperimetric-type inequalities in the plane.

Abstract

The object of the paper is to find complete systems of inequalities relating the perimeter , the area and the Cheeger constant of planar sets. To do so, we study the so called Blaschke--Santaló diagram of the triplet for different classes of domains: simply connected sets, convex sets and convex polygons with at most sides. We completely determine the diagram in the latter cases except for the class of convex -gons when 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.
Paper Structure (22 sections, 12 theorems, 103 equations, 12 figures)

This paper contains 22 sections, 12 theorems, 103 equations, 12 figures.

Table of Contents

  1. Introduction and main results
  2. Results on the classes of simply connected and convex set
  3. Results on the classes of convex polygons
  4. Classical results and preliminaries
  5. Classical definitions and results
  6. Preliminary lemmas
  7. Proofs of the main results
  8. Proof of the inequality \ref{['eq:new_inequality']}
  9. Proof of the second assertion of Theorem \ref{['th:diagram_open']} (convex sets)
  10. Proof of the first assertion of Theorem \ref{['th:diagram_open']} (simply connected sets)
  11. Proof of the inequality \ref{['eq:uper_bound']}
  12. Proof of Theorem \ref{['th:diagram_polygon']}
  13. If $N=3$
  14. If $N$ is even
  15. If $N\ge 5$ is odd
  16. ...and 7 more sections

Key Result

Theorem 2

We take $x_0=P(B) = 2\sqrt{\pi}$.

Figures (12)

  • Figure 1: The Blaschke--Santaló diagrams for the classes of simply connected sets and convex sets.
  • Figure 2: The Blaschke--Santaló diagrams for the classes of simply connected sets and convex sets represented in $[0,1]^2$.
  • Figure 3: Case 1 on the left and case 2 on the right.
  • Figure 4: Rotation of the free side along its midpoint.
  • Figure 5: The continuous and closed path $\Gamma_p$
  • ...and 7 more figures

Theorems & Definitions (30)

  • Definition 1
  • Theorem 2
  • Definition 3
  • Definition 4
  • Theorem 5
  • Theorem 6
  • Remark 7
  • Definition 8
  • Definition 9
  • Theorem 10
  • ...and 20 more