The monotonicity of the Cheeger constant for parallel bodies

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.