发表机构
Iowa State University; Concordia University（爱荷华州立大学; 康考迪亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究高斯扭转刚度的Brunn--Minkowski型不等式，证明其在任意维数$n\boldsymbol{\u2265}2$及任意指数$\boldsymbol{\u03b1}>0$下均不成立，否定了相关猜想与问题，揭示了Minkowski扰动下的一阶增长机制。
AI 中文摘要
设$u$为$\boldsymbol{R}^n$中有界区域$\boldsymbol{\u03a9}$上Ornstein--Uhlenbeck算子的扭转函数，即满足$\boldsymbol{\u0394}u - \boldsymbol{x} \boldsymbol{\u2022} \boldsymbol{\u2207}u=-1$（在$\boldsymbol{\u03a9}$内）且在$\boldsymbol{\u2202\u03a9}$上$u=0$的解。令$\boldsymbol{\u0394}_\text{g}(\boldsymbol{\u03a9})=\boldsymbol{\u222b}_\boldsymbol{\u03a9} u \boldsymbol{d}\u03b3$为高斯扭转刚度。我们证明，对于原点中心对称的一对凸体（可取为具正曲率的光滑凸体），在任意维数$n\boldsymbol{\u2265}2$及任意指数$\boldsymbol{\u03b1}>0$下，Brunn--Minkowski型不等式$\boldsymbol{\u0394}_\text{g}((1-t)\boldsymbol{\u03a9}_0+t\boldsymbol{\u03a9}_1)^\boldsymbol{\u03b1}\boldsymbol{\u2264}(1-t)\boldsymbol{\u0394}_\text{g}(\boldsymbol{\u03a9}_0)^\boldsymbol{\u03b1}+t\boldsymbol{\u0394}_\text{g}(\boldsymbol{\u03a9}_1)^\boldsymbol{\u03b1}$均不成立，这否定了Marín Sola与Salerno的猜想1.4及问题(Q)。该结论的机制是Minkowski扰动下球$\boldsymbol{\u03a9}_0$的$\boldsymbol{\u0394}_\text{g}$一阶下界：当扰动体$\boldsymbol{\u03a9}_1$具有小扭转及大平均宽度时，$\boldsymbol{\u0394}_\text{g}((1-t)\boldsymbol{\u03a9}_0 + t\boldsymbol{\u03a9}_1)$一阶增长，因Minkowski组合的扭转刚度大于两端，故无指数可修复该不等式。
英文摘要
Let $u$ be the torsion function for the Ornstein-Uhlenbeck operator on a bounded domain $Ω\subset \mathbb{R}^n$, i.e., the solution of $Δu - x \cdot \nabla u = -1$ in $Ω$ with $u = 0$ on $\partialΩ$. Let $T_γ(Ω) = \int_Ωu \, dγ$ be the Gaussian torsional rigidity. We prove that the Brunn-Minkowski-type inequality $T_γ((1-t)Ω_0 + tΩ_1)^α \le (1-t) T_γ(Ω_0)^α + t T_γ(Ω_1)^α$ fails for every exponent $α> 0$ 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 Marín Sola and Salerno in the negative. The mechanism is a first-order lower bound for $T_γ$ at a ball $Ω_0$ under Minkowski perturbations. When the perturbing body $Ω_1$ has small torsion and large mean width, $T_γ((1-t)Ω_0 + tΩ_1)$ 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.
CommentsIn 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
