Talagrand高斯凸化问题的一个构造性解
A constructive solution to Talagrand's Gaussian convexification problem
- Institute Louis Bachelier — Fondation du Risque（路易·巴舍利耶研究所—风险基金会）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对Talagrand高斯凸化问题，提出有限维构造方法，利用分离论证、高斯可积性和有限网，构造满足概率下界的大凸子集及最优尺度椭球。
AI中文摘要：
Talagrand 提出了一个问题：在一个有界闵可夫斯基和的大高斯集中，显式构造一个大的凸子集。设 $A\subset\mathbb{R}^n$ 为闭集且 $\gamma_n(A)\ge 7/8$。我们证明，对于每个 $\lambda>4$ 和每个 $4\le t<\lambda$，可以构造一个有限多面体 $C\subset\mathbb{R}^n$，使得 \\[ C\subset \lambda(A+A+A) \qquad\text{且}\qquad \gamma_n(C)\ge 1-\frac1t. \\] 该构造是有限维的：一个分离论证产生多面体凸障碍，高斯可积性给出均匀斜率界，一个有限网将全局问题归结为有限凸最小化问题。作为应用，我们给出了一个最优尺度椭球的有限半定构造。存在一个普适常数 $c>0$，使得对于 $n\ge2$，可以构造一个椭球 $E$，满足 \\[ \gamma_n(E)\ge\frac12 \qquad\text{且}\qquad c\sqrt{\frac{\log n}{n}}\\,E\subset \lambda(A+A+A). \\] 同样的有限障碍方法也为平衡高斯集产生一个构造性的有界步凸化结果。
英文摘要：
Talagrand asked for a construction of a large convex subset of a bounded Minkowski sum of a large Gaussian set. We first prove a stronger nonsymmetric existential statement: if $A\subset\mathbb{R}^n$ is measurable and $γ_n(A)>2/3$, then $A+A+A$ contains a compact convex set of Gaussian measure at least $1/2$. Let now $A$ be closed with $γ_n(A)\ge7/8$, let $Φ$ be the standard Gaussian distribution function, and put $a_0=Φ^{-1}(3/4)$. For every $Λ>1$ and every $0<p<2Φ(Λa_0)-1$ we construct a centrally symmetric finite polytope $C\subsetΛ(A+A+A)$ with $γ_n(C)\ge p$. Thus measure $3/4$ is obtained for every $Λ>1.705510\ldots$. We give matching upper and lower bounds for the optimal high-measure dilation and show that the dilation profile used by the construction is sharp among all symmetric half-measure cores. Finally, if $γ_n(A)\ge5/6+η$, we construct, for every $0<\varepsilon\le1/2$, a centered ellipsoid $E_\varepsilon$ with $γ_n(E_\varepsilon)\ge1-\varepsilon$ and \begin{equation} \frac{c}{Φ^{-1}(1-\varepsilon/2)}\sqrt{\frac{\log n}{n}}\,E_\varepsilon\subset A+A+A. \end{equation} Both the dimension dependence and the dependence on $\varepsilon$ are optimal up to constants. The construction also gives a six-summand theorem for balanced sets and a second finite construction based on subgaussian tests.
