有限维球交集的Gelfand宽度估计

各向异性有限维球之交的Kolmogorov宽度：情形 $2\le q_j<\infty$

Kolmogorov widths of an intersection of anisotropic finite-dimensional balls: case $2\le q_j<\infty$

本文在各向异性有限维空间中，针对指标满足 $2\le q_j<\infty$ 的情形，给出了若干加权球之交的 Kolmogorov $n$-宽度的阶估计。

$1\le q\le 2$ 时 $l_q^k$ 中各向异性有限维球交集的 Kolmogorov 宽度

Kolmogorov widths of an intersection of anisotropic finite-dimensional balls in $l_q^k$ for $1\le q\le 2$

该研究针对 $l_q^k$ 空间中任意一族各向异性范数单位球的交集，在 $1\le q\le 2$ 且 $n\le k/2$ 时建立了其 Kolmogorov $n$-宽度的阶估计。

Ball Intersection Properties in Metric Spaces

We show that in complete metric spaces, $4$-hyperconvexity is equivalent to finite hyperconvexity. Moreover, every complete, almost $n$-hyperconvex metric space is $n$-hyperconvex. This generalizes among others results of Lindenstrauss and answers questions of Aronszajn-Panitchpakdi. Furthermore, we prove local-to-global results for externally and weakly externally hyperconvex subsets of hyperconvex metric spaces and find sufficient conditions in order for those classes of subsets to be convex with respect to a geodesic bicombing.

关于经典与 Schatten $p$-球临界相交的一点注记

A note on critical intersections of classical and Schatten $p$-balls

本文研究了 $\mathbb{R}^n$ 中两个范数单位球相交的渐近体积，提供了通用框架并特化至经典 $\ell_p^n$-球（$p=\infty$）与 Schatten $p$-球（$p=2,\infty$）情形，揭示了非常规的极限行为。

Intersection of unit balls in classical matrix ensembles

We study the volume of the intersection of two unit balls from one of the classical matrix ensembles GOE, GUE and GSE, as the dimension tends to infinity. This can be regarded as a matrix analogue of a result of Schechtman and Schmuckenschläger for classical $\ell_p$-balls [Schechtman and Schmuckenschläger, GAFA Lecture Notes, 1991]. The proof of our result is based on two ingredients, which are of independent interest. The first one is a weak law of large numbers for a point chosen uniformly at random in the unit ball of such a matrix ensemble. The second one is an explicit computation of the…

关于两个$L_p^n$球相交的体积

On the volume of the intersection of two $L_p^n$ balls

本说明涉及以下问题，$p=1$，$q=2$的情况下，其中介绍给我们的维塔利米尔曼：什么是体积剩下的$L_p^n$球后，删除t倍的$L_q^n$球？回想一下，$L_r^n$球是集合$\{(t_1,t_2,\dots,t_n);\ t_i\in{\bf R},\ n^{-1}\sum_{i=1}^n|t_i|^r\le 1\}$，并注意对于$0<p<q<\infty$，$L_q^n$球包含在$L_p^n$球中。在推论4中，我们证明了，在归一化勒贝格测度使得$L_p^n$球的体积为1之后，上述问题的答案对于$T<t<{1\over 2}n^ {{1\over p}-{1\over q}}$是$e^{-ct^pn^{p/q}}$阶的，其中$c$和$T$依赖于$p$和$q$，而不依赖于$n$。主要定理，定理3，处理相应的问题，为表面测量的$L_p^n$球。

Zong问题：包含单位球的$2d$个半空间

Zong's problem on $2d$ half-spaces containing the unit ball

本文证明包含单位球的$2d$个半空间交集必含范数不小于$\nsqrt{d}$的点，等号仅对立方体成立，并给出基于高斯象限概率的证明及扩展。

Limit theorems for the volumes of small codimensional random sections of $\ell_p^n$-balls

We establish Central Limit Theorems for the volumes of intersections of $B_{p}^n$ (the unit ball of $\ell_p^n$) with uniform random subspaces of codimension $d$ for fixed $d$ and $n\to \infty$. As a corollary we obtain higher order approximations for expected volumes, refining previous results by Koldobsky and Lifschitz and approximations obtained from the Eldan--Klartag version of CLT for convex bodies. We also obtain a Central Limit Theorem for the Minkowski functional of the intersection body of $B_p^n$, evaluated on a random vector distributed uniformly on the unit sphere.

关于Schmuckenschlager特征值不等式

On eigenvalue inequalities of Schmuckenschlager

本文重新审视Schmuckenschläger关于两个球交集Dirichlet拉普拉斯最低特征值不等式，提出一种直接且初等的无Lieb证明方法。

高维Orlicz球中的尖锐集中现象

Sharp concentration phenomena in high-dimensional Orlicz balls

本文针对高维Orlicz球的交集给出精确偏差公式，并建立临界情形下的中心极限定理、强大数定律及薄壳集中公式，方法结合调和分析与指数测度变换。

High-dimensional limit theorems for random vectors in $\ell_p^n$-balls

In this paper, we prove a multivariate central limit theorem for $\ell_q$-norms of high-dimensional random vectors that are chosen uniformly at random in an $\ell_p^n$-ball. As a consequence, we provide several applications on the intersections of $\ell_p^n$-balls in the flavor of Schechtman and Schmuckenschläger and obtain a central limit theorem for the length of a projection of an $\ell_p^n$-ball onto a line spanned by a random direction $θ\in\mathbb S^{n-1}$. The latter generalizes results obtained for the cube by Paouris, Pivovarov and Zinn and by Kabluchko, Litvak and Zaporozhets. Moreov…