On the Cheeger inequality for convex sets

Ilias Ftouhi (1)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

In this paper, we prove new sharp bounds for the Cheeger constant of planar convex sets that we use to study the relations between the Cheeger constant and the first eigenvalue of the Laplace operator with Dirichlet boundary conditions. This problem is closely related to the study of the so-called Cheeger inequality for which we provide a significant improvement in the class of planar convex sets, which allows to extend an existence result of [25] to dimension 3. We finally, provide some new sharp bounds for the first Dirichlet eigenvalue of planar convex sets and a new sharp upper bound for triangles that is better than the conjecture stated in [30] in the case of thin triangles.

Domaines: Mathématiques [math]

Ilias Ftouhi. On the Cheeger inequality for convex sets. Journal of Mathematical Analysis and Applications, 2021. ⟨hal-03006015v5⟩