任意测度的同构Busemann-Petty问题：最优阶
Isomorphic Busemann--Petty for arbitrary measures: the sharp order
浏览论文内容
中文总结 AI 辅助
该研究针对任意测度的同构Busemann-Petty问题，证明了最优常数$C_n$的阶为$\boldsymbol{\boldsymbol{\rho}}$，通过结合前期构造与随机舍入等方法得到了匹配的上下界。
中文摘要 AI 辅助
设$C_n$为满足下述性质的最优常数：对$\boldsymbol{R}^n$上任意偶的、连续的、严格正的密度函数$f$，以及所有关于原点对称的凸体$K,L\boldsymbol{\boldsymbol{R}}^n$，若对所有$\boldsymbol{\boldsymbol{R}}^{n-1}$中的$\boldsymbol{\boldsymbol{\rho}}$，都有$\boldsymbol{\boldsymbol{\rho}}$方向上的积分$\boldsymbol{\boldsymbol{\rho}}$与$\boldsymbol{\boldsymbol{\rho}}^\boldsymbol{\boldsymbol{\rho}}$的交集上$f$的积分不超过$L$与$\boldsymbol{\boldsymbol{\rho}}^\boldsymbol{\boldsymbol{\rho}}$的交集上$f$的积分，则$K$上$f$的积分不超过$C_n$乘以$L$上$f$的积分。在之前的论文中，作者证明了$C_n \boldsymbol{\boldsymbol{\rho}}$，本文证明了匹配的下界$C_n \boldsymbol{\boldsymbol{\rho}}$。为简化论述，作者首先基于Klartag和Koldobsky的前期工作给出了完整的单尺度构造，得到$C_n \boldsymbol{\boldsymbol{\rho}}$；对于最优结果，作者将Klartag和Livshyts的随机舍入构造作为黑箱，结合球平均支撑分离论证。
英文摘要
Let $C_n$ be the optimal constant with the following property. For every even, continuous, strictly positive density $f$ on $R^n$ and all origin-symmetric convex bodies $K,L\subset R^n$, the inequalities $$ \int_{K\capξ^\perp}f \leq \int_{L\capξ^\perp}f \qquad\text{for all }ξ\in S^{n-1} $$ imply $\int_Kf\leq C_n\int_Lf$. In an earlier paper the authors proved that $C_n\leq\sqrt n$. In this paper, we prove the matching lower bound $C_n\geq c\sqrt n$. To simplify the exposition, we first give a complete one-scale construction, based on earlier work of Klartag and Koldobsky, which yields $C_n\geq c\sqrt{n/\log n}$. For the sharp result, we use the random-rounding construction of Klartag and Livshyts as a black box and combine it with a spherical-averaging support-separation argument.
