3 problems

Obsolescence conjecture. Condition 1 in Theorem 4.1 is obsolete in general, meaning that the theorem should remain valid without this condition.

Let $T_1$ and $T_2$ be triangles, and write $\lambda_i(T_j)$ for their $i$th eigenvalues of the Dirichlet Laplacian. Antunes–Freitas conjecture. If … then $T_1$ and $T_2$ are ident…

Let $n\geq 3$, let $\Omega$ be a convex $n$-gon, and let $P$ be a regular $n$-gon. Write $\lambda_i(\Omega)$ for the $i$th eigenvalue of the Dirichlet Laplacian on $\Omega$, and le…