Uniform stability of the ball with respect to the first Dirichlet and Neumann infty-eigenvalues

In this note we analyze how perturbations of a ball $\mathfrak{B}_r \subset \mathbb{R}^n$ behaves in terms of their first (non-trivial) Neumann and Dirichlet $\infty-$eigenvalues when a volume constraint $\\mathscr{L}^n(\Omega) = \mathscr{L}^n(\mathfrak{B}_r)$ is imposed. Our main result states that $\Omega$ is uniformly close to a ball when it has first Neumann and Dirichlet eigenvalues close to the ones for the ball of the same volume $\mathfrak{B}_r$. In fact, we show that, if $$ |\lambda_{1,\infty}^D(\Omega) - \lambda_{1,\infty}^D(\mathfrak{B}_r)| = \delta_1 \quad \text{and} \quad |\lambda_{1,\infty}^N(\Omega) - \lambda_{1,\infty}^N(\mathfrak{B}_r)| = \delta_2, $$ then there are two balls such that $$\mathfrak{B}_{\frac{r}{\delta_1 r+1}} \subset \Omega \subset \mathfrak{B}_{\frac{r+\delta_2 r}{1-\delta_2 r}}.$$ In addition, we also obtain a result concerning stability of the Dirichlet $\infty-$eigen-functions.