AI 中文总结
该研究探讨高维凸几何中克拉默变换与半空间深度的关系，证明相关尖锐比较式，推导矩、尾估计及随机凸包阈值准则，建立大偏差代价等的定量联系。
AI 中文摘要
我们研究对数凹概率测度的克拉默变换（Cramér transform）与图基半空间深度（Tukey's half-space depth）之间的关系。对于凸体$K\ni\boldsymbol{R}^n$上的均匀概率测度$\boldsymbol{\nu}_K$，我们证明了内部点$\boldsymbol{x}\ni\boldsymbol{int}(K)$处的逐点尖锐比较：$\boldsymbol{\nu}_K^*(\boldsymbol{x})\ni-\boldsymbol{\nu}_K(\boldsymbol{x})\ni\boldsymbol{\nu}_K^*(\boldsymbol{x})+\frac{1}{2}\boldsymbol{\nu}_K n+C$，其中$C$为绝对常数。该$\boldsymbol{\nu}_K n$阶是最优的，欧氏球可验证这一点。证明结合了指数倾斜、一维对数凹性及克拉默变换的自协调性。作为推论，我们得到$\boldsymbol{\nu}_K^*$的尖锐阶矩与尾估计，并确定$\boldsymbol{\nu}_K^*(\boldsymbol{x})$（相差维度的多项式因子）为$x$被随机凸包捕获所需的独立样本数。我们还建立了对数半空间深度的$\boldsymbol{O}(n^2)$方差界，并用其推导随机凸包尖锐阈值的一般准则，该准则适用于所有$p>1$的$\boldsymbol{\nu}_p$-球上的均匀测度。这些结果在高维凸几何中建立了大偏差代价、几何深度与采样复杂度之间的定量联系。
英文摘要
We study the relationship between the Cramér transform and Tukey's half-space depth for log-concave probability measures. For the uniform probability measure $μ_K$ on a convex body $K\subseteq\mathbb{R}^n$, we prove the sharp pointwise comparison $$Λ_K^*(x)\leq -\log q_K(x)\leq Λ_K^*(x)+\frac12\log n+C,\qquad x\in\operatorname{int}(K),$$ where $C$ is an absolute constant. The order $\log n$ is optimal, as shown by the Euclidean ball. The proof combines exponential tilting, one-dimensional log-concavity, and self-concordance of the Cramér transform. As consequences, we obtain sharp-order moment and tail estimates for $Λ_K^*$ and identify $\exp(Λ_K^*(x))$, up to polynomial factors in the dimension, with the number of independent samples needed for $x$ to be captured by their random convex hull. We also establish an $O(n^2)$ variance bound for the logarithmic half-space depth and use it to derive a general criterion for sharp thresholds of random convex hulls. In particular, this criterion applies to the uniform measures on $\ell_p$-balls for every $p>1$. These results establish a quantitative link between large-deviation cost, geometric depth, and sampling complexity in high-dimensional convex geometry.
