薄壳性质通过高斯倾斜蕴含小球偏差
Thin-Shell implies small-ball deviation via Gaussian tilts
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中文总结 AI 辅助
该研究针对ℝⁿ上的各向同性对数凹测度，利用中心高斯倾斜方法将薄壳性质与小球偏差估计关联，改进了偏差估计的依赖关系，进而证明薄壳性质蕴含切片性质。
中文摘要 AI 辅助
我们证明，对于ℝⁿ上的（偶）各向同性对数凹测度μ，一致薄壳估计能给出欧氏范数|X|低于期望的精确显式偏差估计，将Klartag-Lehec的平方根依赖改进为二次依赖（在数值常数范围内是最优的）。利用近期Chen-Klartag的尖锐方差界，我们推导出：对所有s∈(0,√n)，有μ(|X|≤√n−s)≤exp{−s²/4 (1+O(s/√n))}。特别地，这给出了一个新的、更清晰的证明：薄壳性质蕴含切片性质（通过小球估计实现）。我们的方法基于使用Brazitikos近期引入的中心高斯倾斜，可视为Eldan随机局部化的确定性版本。
英文摘要
We show that uniform thin-shell estimates for isotropic log-concave measures $μ$ on $\mathbb{R}^n$ yield precise and explicit deviation estimates for the Euclidean norm $|X|$ below the expectation, improving the square-root dependence of Klartag-Lehec to a quadratic one (which is best possible, up to numeric constants). Using the recent Chen-Klartag sharp variance bound, we deduce: $$ μ\left(|X|\le \sqrt{n}-s\right) \le \exp\left\{ -\frac{s^2}{4} \left(1+O\left(\frac{s}{\sqrt{n}}\right)\right) \right\} \qquad \forall\, s\in(0,\sqrt{n}). $$ In particular, this yields a new and transparent proof that Thin-Shell implies Slicing (by passing through small-ball estimates). Our method is based on using central Gaussian tilts, recently introduced by Brazitikos, which may be thought of as a deterministic version of Eldan's stochastic localization. An anisotropic variant (when $μ$ has general covariance structure) of these deviation estimates is also obtained.
发表机构
- University of Crete（克里特大学）
- Technion – Israel Institute of Technology（以色列理工学院）
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