The equality case in a Poincar\'e-Wirtinger type inequality
  In this paper, generalizing to the non smooth case already existing results, we prove that, for any convex planar set $\Omega$, the first non-trivial Neumann eigenvalue $\mu_1(\Omega)$ of the Hermite operator is greater than or equal to 1. Furthermore, and this is our main result, under some additional assumptions on $\Omega$, we show that $\mu_1(\Omega)=1$ if and only if $\Omega$ is any strip. The study of the equality case requires, among other things, an asymptotic analysis of the eigenvalues of the Hermite operator in thin domains. 
B. Brandolini, F. Chiacchio, D. Krej\v{c}i\v{r}\'ik, C. Trombetti
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