AI 中文总结
该研究针对高维随机多面体的期望面数问题，推导了乘积测度下的下界、对数凹测度下的普适标度界，并构造非对数凹反例，揭示了测度性质对随机多面体面数的影响。
AI 中文摘要
我们证明了高维随机多面体的期望面数具有阶为$n^{n/2}e^{O(n)}$的界。首先，设$μ$是$\boldsymbol{\rm R}$上非退化、具有紧支撑的偶概率测度，且在其右端点$x^\text{*}$附近满足$μ([x^\text{*}-s,x^\text{*}])\text{asymp}s^κ$。对任意充分小的固定$α>0$，服从分布$μ^{\text{⊗}n}$的$N=\text{⌊}e^{αn}\text{⌋}$个独立点的凸包，其期望面数至少为$n^{n/2}e^{-C_{μ,α}n}$；这一结论涵盖所有对称有限字母表分布。对于$\boldsymbol{\rm R}^n$上的任意满维对数凹概率测度，我们证明存在$T∈[n,2n]$和$N=\text{⌈}e^Tn^{3/2}\text{⌉}$，使得$n^{n/2}e^{-Cn} ≤ \boldsymbol{\rm E} f_{n-1}(P_N) ≤ n^{n/2}e^{Cn}$。因此在高维指数区域中，不计指数因子的话，$n^{n/2}$这一标度对对数凹测度是普适的。最后，我们构造了一个对称迷向、满支撑的非对数凹反例，其期望面数仅为$(1+o(1))2^n$。
英文摘要
We prove bounds of order $n^{n/2}e^{O(n)}$ for the expected number of facets of high-dimensional random polytopes. First, let $μ$ be a non-degenerate compactly supported even probability measure on $\R$ satisfying $μ([x^\ast-s,x^\ast])\asymp s^κ$ near its right endpoint $x^\ast$. For every sufficiently small fixed $α>0$, the convex hull of $N=\lfloor e^{αn}\rfloor$ independent points with law $μ^{\otimes n}$ has at least $n^{n/2}e^{-C_{μ,α}n}$ expected facets; this includes all symmetric finite-alphabet distributions. For every full-dimensional log-concave probability measure on $\R^n$, we prove that there exist $T\in[n,2n]$ and $N=\lceil e^Tn^{3/2}\rceil$ for which \[ n^{n/2}e^{-Cn} \leq \mathbb E f_{n-1}(P_N) \leq n^{n/2}e^{Cn}. \] Thus the scale $n^{n/2}$, up to exponential factors, is universal for log-concave measures in this high-dimensional exponential regime. Finally, we construct a symmetric isotropic full-support non-log-concave counterexample with only $(1+o(1))2^n$ expected facets.
