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 $T_{\gamma }(\Omega )={\int }_{\Omega }ud\gamma$ 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 $n\ge 2$, 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 ${\Omega }_0$ 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.

arXiv commentIn our first version of the manuscript, we misquoted Marín Sola and Salerno. We thank them for pointing it out. Their conjecture 1.4 is stated for all dimensions $n$ and the Question (Q) is for $n\in \{1,2\}$. After their communication, we addressed these points and added the section about dimension one