A reverse quantitative isoperimetric type inequality for the Dirichlet Laplacian

A stability result in terms of the perimeter is obtained for the first Dirichlet eigenvalue of the Laplacian operator. In particular, we prove that, once we fix the dimension $n\ge 2$, there exists a constant $c>0$, depending only on $n$, such that, for every $\mathrm{\Omega }\subset {\mathbb{R}}^n$ open, bounded and convex set with volume equal to the volume of a ball $B$ with radius $1$, it holds \begin{equation*} \lambda_1(\Omega)-\lambda_1(B)\geq c\left(P(\Omega)-P(B) \right)^{2}, \end{equation*} where by ${\lambda }_1(\cdot )$ we denote the first Dirichlet eigenvalue of a set and by $P(\cdot )$ its perimeter. The hearth of the present paper is a sharp estimate of the Fraenkel asymmetry in terms of the perimeter.