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 $T_{\gamma }^{1\mathrm{/}(n+2)}$ is neither convex nor concave. On the positive side, we prove that $T_{\gamma }^{1\mathrm{/}3}$ 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 ${\lambda }_{\gamma }({\mathrm{\Omega }}_t{)}^{-1\mathrm{/}2}\ge (1-t){\lambda }_{\gamma }({\mathrm{\Omega }}_0{)}^{-1\mathrm{/}2}+t{\lambda }_{\gamma }({\mathrm{\Omega }}_1{)}^{-1\mathrm{/}2}$ does not hold, even for centrally symmetric sets.