Remarks on Brunn-Minkowski-type inequalities related to the Ornstein-Uhlenbeck operator

We investigate Brunn-Minkowski-type inequalities for the torsional rigidity $T_\gamma$ and the first eigenvalue $\lambda_\gamma$ associated with the Ornstein-Uhlenbeck operator. Counterexamples are provided showing that neither concavity nor convexity properties hold for $T_\gamma$ on general bounded convex sets. We also demonstrate that log-concavity and log-convexity properties fail in this setting. In the case of centrally symmetric sets, we answer a question raised by Cordero-Erausquin and Eskenazis by showing that $T_\gamma^{1/(n+2)}$ is neither convex nor concave. On the positive side, we prove that $T_\gamma^{1/3}$ is convex with respect to Minkowski addition when restricted to Euclidean balls centered at the origin. For $\lambda_\gamma$, we answer negatively a question posed by Colesanti, Francini, Livshyts, and Salani by showing that the inequality $\lambda_\gamma(\Omega_t)^{-1/2} \geq (1-t)\lambda_\gamma(\Omega_0)^{-1/2} + t\lambda_\gamma(\Omega_1)^{-1/2}$ does not hold, even for centrally symmetric sets.