On the Cheeger inequality for convex sets

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 an improvement in the class of planar convex sets. We then provide an existence theorem that highlights the tight relation between improving the Cheeger inequality and proving the existence of a minimizer of the functional J:=λ₁/h² in any dimension n. We finally, provide some new sharp bounds for the first Dirichlet eigenvalue of planar convex sets and a new sharp upper bound for triangles which is better than the conjecture stated in [32] in the case of thin triangles.