Poincaré型不等式中的交换性
Commutativity in a Poincaré-Type Inequality
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中文总结 AI 辅助
本文研究高斯测度在原点对称凸集上的Poincaré型不等式，证明其成立等价于矩阵交换性，并给出非交换情形下的反例及无条件凸集上的充分条件。
中文摘要 AI 辅助
我们研究了限制在原点对称凸集上的高斯测度的Poincaré型不等式。该不等式源于函数$t\longmapsto \gamma_{\Sigma}\left(e^{tA}K\right)$在$t=0$处的二阶变分，其中$\Sigma$和$A$是对称矩阵。该函数的相关性来自Saroglou对log-Brunn-Minkowski猜想的重新表述。我们证明$\Sigma$和$A$交换当且仅当该不等式对每个形如$UD B_\infty^n$的盒子成立，其中$U$是正交矩阵，$D$是对角矩阵。在非交换情形下，这为正交盒子提供了反例，其轴可以任意接近坐标轴。我们还证明了当$K$关于$\Sigma$的特征基是无条件且$A$是任意对称矩阵时，该不等式成立。
英文摘要
We study a Poincaré-type inequality for Gaussian measures restricted to origin-symmetric convex sets. The inequality arises from the second variation at $t=0$ of the function $t\longmapsto γ_Σ\left(e^{tA}K\right)$, where $Σ$ and $A$ are symmetric matrices. The relevance of this function comes from Saroglou's reformulation of the log-Brunn--Minkowski conjecture. We show that $Σ$ and $A$ commute if and only if the inequality holds for every box of the form $UD B_\infty^n$, where $U$ is orthogonal and $D$ is diagonal. In the non-commuting case, this yields counterexamples for orthogonal boxes whose axes can be chosen arbitrarily close to the coordinate axes. We also show that the inequality holds when $K$ is unconditional with respect to an eigenbasis of $Σ$ and $A$ is an arbitrary symmetric matrix.
