Kolmogorov widths of the intersection of two finite-dimensional balls

混合范数中球的Kolmogorov宽度：刚性情形

Kolmogorov widths of balls in mixed norms: the case of rigidity

本文刻画了混合范数空间中球为刚性的参数条件，即大维数线性子空间逼近效果差，结合已有下界与新构造解决了宽度估计的关键定性情形。

$1\le q\le 2$ 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。

有限维球交集的Gelfand宽度估计

Estimates for the Gelfand widths of intersections of finite-dimensional balls

该研究在特定参数条件下，推导得出有限维球交集的Gelfand宽度的阶估计结果，属于泛函分析领域的理论研究。

Kolmogorov and linear widths of the weighted Besov classes with singularity at the origin

In this paper we obtain asymptotic estimates of Kolmogorov and linear widths of the weighted Besov classes with singularity at the origin. In addition, estimates of Kolmogorov and linear widths of finite-dimensional balls in a mixed norm are obtained.

各向异性 Sobolev 类的 Kolmogorov 宽度

Kolmogorov widths of anisotropic Sobolev classes

本文研究了周期各向异性 Sobolev 和 Nikol'skii 类以及各向异性有限维球的 Kolmogorov 宽度，获得了相应的阶估计。

Kolmogorov widths on the sphere via eigenvalue estimates for Hölderian integral operators

Approximation processes in the reproducing kernel Hilbert space associated to a continuous kernel on the unit sphere $S^m$ in the Euclidean space $\mathbb{R}^{m+1}$ are known to depend upon the Mercer's expansion of the compact and self-adjoint $L^2(S^m)$-operator associated to the kernel. The estimation of the Kolmogorov $n$-th width of the unit ball of the reproducing kernel Hilbert space in $L^2(S^m)$ and the identification of the so-called optimal subspace usually suffice. These Kolmogorov widths can be computed through the eigenvalues of the integral operator associated to the kernel. Thi…

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…

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

This note deals with the following problem, the case $p=1$, $q=2$ of which was introduced to us by Vitali Milman: What is the volume left in the $L_p^n$ ball after removing a t-multiple of the $L_q^n$ ball? Recall that the $L_r^n$ ball is the set $\{(t_1,t_2,\dots,t_n);\ t_i\in{\bf R},\ n^{-1}\sum_{i=1}^n|t_i|^r\le 1\}$ and note that for $0<p<q<\infty$ the $L_q^n$ ball is contained in the $L_p^n$ ball. In Corollary 4 we show that, after normalizing Lebesgue measure so that the volume of the $L_p^n$ ball is one, the answer to the problem above is of order $e^{-ct^pn^{p/q}}$ for $T<t<{1\over 2}n…

Isomorphic properties of Intersection bodies

We study isomorphic properties of two generalizations of intersection bodies, the class of k-intersection bodies and the class of generalized k-intersection bodies. We also show that the Banach-Mazur distance of the k-intersection body of a convex body, when it exists and it is convex, with the Euclidean ball, is bounded by a constant depending only on k, generalizing a well-known result of Hensley and Borell. We conclude by giving some volumetric estimates for k-intersection bodies.

Bounds on Kolmogorov widths and sampling recovery for classes with small mixed smoothness

Results on asymptotic characteristics of classes of functions with mixed smoothness are obtained in the paper. Our main interest is in estimating the Kolmogorov widths of classes with small mixed smoothness. We prove the corresponding bounds for the unit balls of the trigonometric polynomials with frequencies from a hyperbolic cross. We demonstrate how our results on the Kolmogorov widths imply new upper bounds for the optimal sampling recovery in the $L_2$ norm of functions with small mixed smoothness.

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.

On the Gaussian measure of the intersection of symmetric, convex sets

The Gaussian Correlation Conjecture states that for any two symmetric, convex sets in n-dimensional space and for any centered, Gaussian measure on that space, the measure of the intersection is greater than or equal to the product of the measures. In this paper we obtain several results which substantiate this conjecture. For example, in the standard Gaussian case, we show there is a positive constant, c, such that the conjecture is true if the two sets are in the Euclidean ball of radius $c\sqrt{n}$. Further we show that if for every n the conjecture is true when the sets are in the Euclidea…

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。

无条件集与随机向量的宽度和刚性

Widths and rigidity of unconditional sets and random vectors

该研究证明坐标循环移位不变的无条件集在ℓ_q^N空间中的刚性，将Gluskin方法用于随机向量平均Kolmogorov宽度研究，通过双正交随机向量弱矩得通用下界，还给出混合范数球宽度下界等推论。

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…

The $ε$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors

We prove an estimation of the Kolmogorov $ε$-entropy in H of the unitary ball in the space V, where H is a Hilbert space and V is a Sobolev-like subspace of H. Then, by means of Zelik's result [5], an estimate of the fractal dimension of the attractors of some nonlinear parabolic equations is established.

与数学物理方程相关的某些奇异类的最优逼近和Kolmogorov宽度估计

Optimal approximation and Kolmogorov widths estimates for certain singular classes related to equations of mathematical physics

针对定义在超立方体上的两类奇异函数类，估计其Kolmogorov和Babenko宽度，并构造局部样条实现最优阶逼近。