The monotonicity of the Cheeger constant for parallel bodies

TL;DR

This paper proves the monotonicity of a specific Cheeger constant-related function for convex sets in the plane, explores its differentiability in any dimension, and derives estimates for Cheeger sets, with potential extensions to other functionals.

Contribution

It establishes the monotonicity of the Cheeger constant function for convex bodies, proves its differentiability in any dimension, and provides estimates on Cheeger sets, extending understanding of geometric inequalities.

Findings

The function involving the Cheeger constant is decreasing for convex sets.

Differentiability of the Cheeger constant map is established in any dimension.

Explicit formulas for derivatives of the Cheeger constant are derived.

Abstract

We prove that for every planar convex set $\Omega$, the function $t\in (-r(\Omega),+\infty)\longmapsto \sqrt{|\Omega_t|}h(\Omega_t)$ is monotonically decreasing, where $r$, $|\cdot|$ and $h$ stand for the inradius, the measure and the Cheeger constant and $(\Omega_t)$ for parallel bodies of $\Omega$. The result is shown to not hold when the convexity assumption is dropped. We also prove the differentiability of the map $t\longmapsto h(\Omega_t)$ in any dimension and without any regularity assumption on $\Omega$, 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.