In this talk, we prove some new sharp bounds for the Cheeger constant of planar convex sets in terms of simple geometric quantities. Such estimates are then used to study the relations between the Cheeger constant and the first eigenvalue of the Laplace operator with Dirichlet boundary condition. This problem is closely related to the study of the classic Cheeger inequality for which we provide an improvement in the class of planar convex sets. The talk is based on joint works with Alba Lia Masiello and Gloria Paoli (Universita degli Studi di Napoli Federico II).
Idriss Mazari
