Reverse Cheeger inequality for planar convex sets

Enea Parini

We prove the sharp inequality \[ J(Ω) := \frac{λ_1(Ω)}{h_1(Ω)^2} < \frac{π^2}{4},\] where $\Omega$ is any planar, convex set, ${\lambda }_1(\Omega )$ is the first eigenvalue of the Laplacian under Dir…