Failure of a Brunn-Minkowski-type inequality for the Gaussian torsional rigidity

Let $u$ be the torsion function for the Ornstein-Uhlenbeck operator on a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, i.e., the solution of $\mathrm{\Delta }u-x\cdot \mathrm{\nabla }u=-1$ in $\mathrm{\Omega }$ with $u=0$ on $\mathrm{\partial }\mathrm{\Omega }$. Let $T_{\gamma }(\mathrm{\Omega })={\int }_{\mathrm{\Omega }}ud\gamma$ be the Gaussian torsional rigidity. We prove that the Brunn-Minkowski-type inequality $T_{\gamma }((1-t){\mathrm{\Omega }}_0+t{\mathrm{\Omega }}_1{)}^{\alpha }\le (1-t)T_{\gamma }({\mathrm{\Omega }}_0{)}^{\alpha }+t{T}_{\gamma }({\mathrm{\Omega }}_1{)}^{\alpha }$ fails for every exponent $\alpha >0$ and in every dimension $n\ge 2$, for a pair of convex bodies centrally symmetric with respect to the origin, which may be taken smooth with positive curvature. This answers Conjecture 1.4 for any $n\ge 2$, and Question~(Q), of Marín Sola and Salerno in the negative. The mechanism is a first-order lower bound for $T_{\gamma }$ at a ball ${\mathrm{\Omega }}_0$ under Minkowski perturbations. When the perturbing body ${\mathrm{\Omega }}_1$ has small torsion and large mean width, $T_{\gamma }((1-t){\mathrm{\Omega }}_0+t{\mathrm{\Omega }}_1)$ increases to first order. Since the Minkowski combination has larger torsion than both endpoints, no exponent can repair the inequality. For $n=1$, convexity with the optimal exponent $1\mathrm{/}3$ holds on symmetric intervals by results of the same authors, \cite{MSS26}. We prove that the logarithm of the torsion is neither convex nor concave along Minkowski combinations of symmetric intervals, that no non-zero exponent yields concavity, and that convexity fails for every positive exponent when one set is a union of two intervals or when the sets are reflected off-center intervals.