
Journal of Convex Analysis 24 (2017), No. 1, 107122 Copyright Heldermann Verlag 2017 Reverse Cheeger Inequality for Planar Convex Sets Enea Parini Centre de Mathématiques et Informatique, AixMarseille University, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France enea.parini@univamu.fr [Abstractpdf] We prove the sharp inequality \[ J(\Omega) := \frac{\lambda_1(\Omega)} {h_1(\Omega)^2} < \frac{\pi^2}{4},\] where $\Omega$ is any planar, convex set, $\lambda_1(\Omega)$ is the first eigenvalue of the Laplacian under Dirichlet boundary conditions, and $h_1(\Omega)$ is the Cheeger constant of $\Omega$. The value on the righthand side is optimal, and any sequence of convex sets with fixed volume and diameter tending to infinity is a maximizing sequence. Morever, we discuss the minimization of $J$ in the same class of subsets: we provide a lower bound which improves the generic bound given by Cheeger's inequality, we show the existence of a minimizer, and we give some optimality conditions. Keywords: Cheeger's inequality. MSC: 49Q10 [ Fulltextpdf (176 KB)] for subscribers only. 