The Minimal Gap Between Λ₂(Ω) and Λ_∞(Ω) in a Class of Convex Domains

We consider the minimization problem $$\begin{aligned}\min_{\Omega\in X}\left(\Lambda_2-\Lambda_\infty\right)(\Omega),\end{aligned}$$ where $\Lambda_2(\Omega)$ and $\Lambda_\infty(\Omega)$ are the (square root of the) first eigenvalue of the Laplacian and the first eigenvalue of the $\infty-$Laplacian respectively. $X$ is the class of convex domains with prescribed diameter. We prove existence of a solution, and we provide several geometrical properties of minimizers