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Journal of Convex Analysis 24 (2017), No. 1, 107--122
Copyright Heldermann Verlag 2017

Reverse Cheeger Inequality for Planar Convex Sets

Enea Parini
Centre de Mathématiques et Informatique, Aix-Marseille University, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France


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 right-hand 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

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