# A conjecture on shape optimization for Dirichlet-Laplacian

For a given domain $G$, with sufficiently smooth boundary, in the plane we denote the first two eigenvalues of Dirichlet-Laplacian on $G$ of by $\lambda_1(G)$ and $\lambda_2(G)$.

$\textbf{Open(?) problem. }$Let $D$ be a box and consider the following set: $$A_\alpha=\{\Big(\lambda_1(\Omega),\lambda_2(\Omega)\Big):\,\,\Omega\subset D, |\Omega|\leq \alpha\},$$ where $\alpha>0$ is fixed and $|.|$ denotes the area measure. Show (or give counterexample) that $A$ is convex.
$\textbf{Remark 1.}$ If we mimic this problem for an interval of finite length it is very easy to observe that the corresponding set $A$ is indeed convex.

$\textbf{Remark 2.}$ In this paper M. Ashbaugh and R.Benguria show that this set is convex in the $x$ and the $y$ direction. See also my favorite survey on this topic by A. Henrot and also this paper.

Has been any progress been made towards resolving this problem?