AI 中文总结
该研究针对(B)-定理提出与维度无关的新稳定性估计，推导了相关近最优体的精确性质，并将方法推广至强、函数型(B)-不等式的稳定性分析。
AI 中文摘要
科德罗-埃劳钦、弗拉德利齐和莫雷的(B)-定理指出：若γ是ℝⁿ上的标准高斯测度，K是ℝⁿ中原点对称凸集，s、t为实数，则γ(e^((s+t)/2)K)≥√(γ(e^sK)γ(e^tK))。赫斯科维奇、利夫希茨、罗滕和沃尔贝格证明了该结果的稳定性版本，表明若K的(B)-不等式等式成立的因子为(1+ε)，则K的内半径要么“极大”要么“极小”，界取决于ε和n。我们给出了一种与维度无关的新稳定性估计，还能提供关于(B)-不等式近最优解的更精确信息。具体而言，我们的结果意味着：若γ(e^((s+t)/2)K)≤(1+ε)√(γ(e^sK)γ(e^tK))，则将高斯测度限制在K上得到的概率测度的协方差矩阵的每个主成分，要么至少为1-O(ε)，要么至多为O(ε)，这是最优的。我们的方法可直接推广以得到(B)-不等式的推广形式（即“强”和“函数型”(B)-不等式）的稳定性估计，这些推广形式可归结为ℝⁿ上1-对数凹测度的谱问题。
英文摘要
The (B)-theorem of Cordero-Erausquin, Fradelizi and Maurey states that if $γ$ is the standard Gaussian in $\mathbb R^n$, $K \subset \mathbb R^n$ is an origin-symmetric convex set, and $s, t \in \mathbb R$ then $γ\left(e^{\frac{s + t}{2}} K\right) \ge \sqrt{γ(e^{s} K) γ(e^{t} K)}$. Herscovici, Livshyts, Rotem and Volberg proved a stability version of this result, showing that if one has equality up to a factor $(1 + ε)$ in the (B)-inequality for $K$ then the inradius of $K$ must be either ``very large'' or ``very small,'' where the bounds depend on $ε$ and on $n$. We give a new stability estimate which is dimension-free and also yields more precise information about bodies which are near-optimizers of the (B)-inequality. In particular, our results imply that if $γ\left(e^{\frac{s + t}{2}} K\right) \le (1 + ε) \sqrt{γ(e^{s} K) γ(e^{t} K)}$, then every principal component of the covariance matrix of the probability measure obtained by restricting the Gaussian to $K$ must either be at least $1 - O(ε)$ or at most $O(ε)$, which is sharp. Our method extends immediately to yield stability estimates for generalizations of the (B)-inequality, namely the ``strong'' and ``functional'' (B)-inequalities, which reduce to spectral questions about $1$-log-concave measures on $\mathbb R^n$.
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