Abstract

We prove the sharp inequality J(Ω):=λ1(Ω)h1(Ω)2<π24,J(\Omega):= \frac{\lambda_1(\Omega)} {h_1(\Omega)^2} < \frac{\pi^2}{4}, where Ω\Omega is any planar, convex set, λ1(Ω)\lambda_1(\Omega) is the first eigenvalue of the Laplacian under Dirichlet boundary conditions, and h1(Ω)h_1(\Omega) is the Cheeger constant of Ω\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 JJ 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.

Author information

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Enea Parini

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

enea.parini@univ-amu.fr

Keywords

  • Cheeger's inequality

Mathematics Subject Classification

49Q10

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Suggested citation

E. Parini. “Reverse Cheeger Inequality for Planar Convex Sets.” Journal of Convex Analysis 24 (2017), No. 1, 107–122. https://doi.org/10.68381/jca24009

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Published by Heldermann Verlag, 2017. Rights now held by Banach Press LLC.