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

Abstract

Let $u$ be the torsion function for the Ornstein-Uhlenbeck operator on a bounded domain $\Omega \subset \mathbb{R}^n$, i.e., the solution of $\Delta u - x \cdot \nabla u = -1$ in $\Omega$ with $u = 0$ on $\partial\Omega$. Let $T_\gamma(\Omega) = \int_\Omega u \, d\gamma$ be the Gaussian torsional rigidity. We prove that the Brunn-Minkowski-type inequality $T_\gamma((1-t)\Omega_0 + t\Omega_1)^{\alpha} \le (1-t) T_\gamma(\Omega_0)^{\alpha} + t T_\gamma(\Omega_1)^{\alpha}$ 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 $n \geq 2$, and Question~(Q), of Mar\'in Sola and Salerno in the negative. The mechanism is a first-order lower bound for $T_\gamma$ at a ball $\Omega_0$ under Minkowski perturbations. When the perturbing body $\Omega_1$ has small torsion and large mean width, $T_\gamma((1-t)\Omega_0 + t\Omega_1)$ 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 $1/3$ 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.