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Sharp inequalities involving the Cheeger constant of planar convex sets

Ilias Ftouhi, Alba Lia Masiello, Gloria Paoli

Abstract

We are interested in finding sharp bounds for the Cheeger constant $h$ via different geometrical quantities, namely the area $|\cdot|$, the perimeter $P$, the inradius $r$, the circumradius $R$, the minimal width $ω$ and the diameter $d$. We provide new sharp inequalities between these quantities for planar convex bodies and enounce new conjectures based on numerical simulations. In particular, we completely solve the Blaschke-Santaló diagrams describing all the possible inequalities involving the triplets $(P,h,r)$, $(d,h,r)$ and $(R,h,r)$ and describe some parts of the boundaries of the diagrams of the triplets $(ω,h,d)$, $(ω,h,R)$, $(ω,h,P)$, $(ω,h,|\cdot|)$, $(R,h,d)$ and $(ω,h,r)$.

Sharp inequalities involving the Cheeger constant of planar convex sets

Abstract

We are interested in finding sharp bounds for the Cheeger constant via different geometrical quantities, namely the area , the perimeter , the inradius , the circumradius , the minimal width and the diameter . We provide new sharp inequalities between these quantities for planar convex bodies and enounce new conjectures based on numerical simulations. In particular, we completely solve the Blaschke-Santaló diagrams describing all the possible inequalities involving the triplets , and and describe some parts of the boundaries of the diagrams of the triplets , , , , and .
Paper Structure (24 sections, 24 theorems, 126 equations, 18 figures)

This paper contains 24 sections, 24 theorems, 126 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Classical results ad preliminary lemmas
  4. Extremal sets and their properties
  5. Inequalities relating the Cheeger constant to one geometric quantity
  6. Numerical results and Blaschke--Santaló diagrams
  7. Generation of random convex polygons
  8. About the computation of the functionals
  9. Proof of the main Results
  10. Proof of Theorem \ref{['th:existence']}
  11. Proof of Theorem \ref{['th2']}
  12. The triplet $(P,h,r)$
  13. The triplet $(d,h,r)$
  14. The triplet $(R,h,r)$
  15. Explicit description of the Blaschke-Santaló diagrams
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\Omega\in \mathcal{K}^2$, then the minimization and the maximization shape optimization problems of the Cheeger constant $h(\Omega)$ admit a solution in the classes of sets defined in $(1)-(13)$.

Figures (18)

  • Figure 1: The idea of the proof of Lemma \ref{['lem:main']}.
  • Figure 2: A stadium.
  • Figure 3: A symmetrical spherical slice.
  • Figure 4: A two-cup body.
  • Figure 5: A Yamanouti set.
  • ...and 13 more figures

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Lemma 2.8
  • ...and 54 more