Geodesic
Reverse Cheeger Inequality for Planar Convex Sets
Journal of convex analysis, Tome 24 (2017) no. 1, pp. 107-122
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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.
Classification : 49Q10
Mots-clés : Cheeger's inequality 
@article{JCA_2017_24_1_JCA_2017_24_1_a8,
     author = {E. Parini},
     title = {Reverse {Cheeger} {Inequality} for {Planar} {Convex} {Sets}},
     journal = {Journal of convex analysis},
     pages = {107--122},
     year = {2017},
     volume = {24},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JCA_2017_24_1_JCA_2017_24_1_a8/}
}
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E. Parini. Reverse Cheeger Inequality for Planar Convex Sets. Journal of convex analysis, Tome 24 (2017) no. 1, pp. 107-122. http://geodesic.mathdoc.fr/item/JCA_2017_24_1_JCA_2017_24_1_a8/