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The monotonicity of the Cheeger constant for parallel bodies

Ilias Ftouhi

TL;DR

The paper proves that for planar convex sets $\Omega$, the scale-invariant quantity $t \mapsto \sqrt{|\Omega_t|}\,h(\Omega_t)$ is nonincreasing with respect to the parallel deformation parameter $t$, with a constant-interval behavior and a strictly decreasing regime unless $\Omega$ is a ball. It then establishes a differentiability result for $t \mapsto h(\Omega_t)$ in any dimension, providing an explicit derivative formula in terms of the Cheeger set boundary and curvature. The authors leverage these results to derive bounds on the contact surface between Cheeger sets and the domain boundary, and discuss extensions to other functionals such as the first Dirichlet eigenvalue, including a Brunn–Minkowski framework and conjectures for higher dimensions. They also show that the convexity assumption is crucial by giving a nonconvex counterexample and discuss perturbation properties in the planar setting. Overall, the work links geometric evolution via parallel bodies to spectral and isoperimetric-type functionals, offering tools for Blaschke–Santaló diagrams and potential generalizations.

Abstract

We prove that for every planar convex set $Ω$, the function $t\in (-r(Ω),+\infty)\longmapsto \sqrt{|Ω_t|}h(Ω_t)$ is monotonically decreasing, where $r$, $|\cdot|$ and $h$ stand for the inradius, the measure and the Cheeger constant and $(Ω_t)$ for parallel bodies of $Ω$. The result is shown to not hold when the convexity assumption is dropped. We also prove the differentiability of the map $t\longmapsto h(Ω_t)$ in any dimension and without any regularity assumption on $Ω$, obtaining an explicit formula for the derivative. Those results are then combined to obtain estimates on the contact surface of the Cheeger sets of convex bodies. Finally, potential generalizations to other functionals such as the first eigenvalue of the Dirichlet Laplacian are explored.

The monotonicity of the Cheeger constant for parallel bodies

TL;DR

The paper proves that for planar convex sets , the scale-invariant quantity is nonincreasing with respect to the parallel deformation parameter , with a constant-interval behavior and a strictly decreasing regime unless is a ball. It then establishes a differentiability result for in any dimension, providing an explicit derivative formula in terms of the Cheeger set boundary and curvature. The authors leverage these results to derive bounds on the contact surface between Cheeger sets and the domain boundary, and discuss extensions to other functionals such as the first Dirichlet eigenvalue, including a Brunn–Minkowski framework and conjectures for higher dimensions. They also show that the convexity assumption is crucial by giving a nonconvex counterexample and discuss perturbation properties in the planar setting. Overall, the work links geometric evolution via parallel bodies to spectral and isoperimetric-type functionals, offering tools for Blaschke–Santaló diagrams and potential generalizations.

Abstract

We prove that for every planar convex set , the function is monotonically decreasing, where , and stand for the inradius, the measure and the Cheeger constant and for parallel bodies of . The result is shown to not hold when the convexity assumption is dropped. We also prove the differentiability of the map in any dimension and without any regularity assumption on , obtaining an explicit formula for the derivative. Those results are then combined to obtain estimates on the contact surface of the Cheeger sets of convex bodies. Finally, potential generalizations to other functionals such as the first eigenvalue of the Dirichlet Laplacian are explored.
Paper Structure (13 sections, 11 theorems, 80 equations, 3 figures)

This paper contains 13 sections, 11 theorems, 80 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Proofs of the main results
  4. Proof of Theorem \ref{['th:monotonicity_cheeger']}
  5. Proof of Theorem \ref{['th:derivative_cheeger']}
  6. Discussion and applications
  7. The non convex case
  8. Results on the contact surface of the Cheeger sets
  9. Perturbation results for the Cheeger constant
  10. About other functionals and generalizations
  11. About the first Dirichlet eigenvalue of the Laplacian
  12. Different functionals and generalizations
  13. Conclusion

Key Result

Theorem 1.2

Let $\Omega$ be an element of $\mathcal{K}^2$.

Figures (3)

  • Figure 1: Two convex sets in $\mathbb{R}^2$ and their corresponding form bodies.
  • Figure 2: Estimating the area of inner sets allows us to estimate the Cheeger constant.
  • Figure 3: Tailed domain that provides a counterexample for Theorem \ref{['th:monotonicity_cheeger']} in the non-convex case.

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 3.1
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Remark 4.3
  • Lemma 4.4
  • ...and 19 more