On a reverse Kohler-Jobin inequality

We consider the shape optimization problems for the quantities $\lambda (\mathrm{\Omega })T^q(\mathrm{\Omega })$, where $\mathrm{\Omega }$ varies among open sets of ${\mathbb{R}}^d$ with a prescribed Lebesgue measure. While the characterization of the infimum is completely clear, the same does not happen for the maximization in the case $q>1$. We prove that for $q$ large enough a maximizing domain exists among quasi-open sets and that the ball is optimal among {\it nearly spherical domains}.