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 asymptotic volume of such matrix unit balls, which in turn is based on the theory of logarithmic potentials with external fields.
Order estimates for the Kolmogorov widths of an intersection of two finite-dimensional balls in a mixed norm under some conditions on the parameters are obtained.
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.
In this paper we obtain order estimates for the Kolmogorov widths of the set $B_{p_0}^m\cap νB_{p_1}^m$ in $l_q^m$; here $0\le n\le m/2$, $p_0>p_1$, $q<\infty$.
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…
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.
CLT for $β$-ensembles with Freud weights, application to the KLS conjecture in Schatten balls
KTH Stockholm（皇家理工学院）；Institut de mathématiques de Toulouse, France（图卢兹数学研究所）；Institut de Mathématiques de Toulouse (UMR5219), Université de Toulouse, CNRS（图卢兹数学研究所（UMR5219），图卢兹大学，法国国家科学研究中心；图卢兹第三大学）；UPS
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…
Busemann's intersection inequality gives an upper bound for the volume of the intersection body of a star body in terms of the volume of the body itself. Koldobsky, Paouris, and Zymonopoulou asked if there is a similar result for $k$-intersection bodies. We solve this problem for star bodies that are close to the Euclidean ball in the Banach-Mazur distance. We also improve a bound obtained by Koldobsky, Paouris, and Zymonopoulou for general star bodies in the case when $k$ is proportional to the dimension.
We study the precise asymptotic volume of balls in Orlicz spaces and show that the volume of the intersection of two Orlicz balls undergoes a phase transition when the dimension of the ambient space tends to infinity. This generalizes a result of Schechtman and Schmuckenschläger [GAFA, Lecture notes in Math. 1469 (1991), 174--178] for $\ell_p^d$-balls. As another application, we determine the precise asymptotic volume ratio for $2$-concave Orlicz spaces $\ell_M^d$. Our method rests on ideas from statistical mechanics and large deviations theory, more precisely the maximum entropy or Gibbs prin…
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.
Reflexive cones in Banach spaces are cones with weakly compact intersection with the unit ball. In this paper we study the structure of this class of cones. We investigate the relations between the notion of reflexive cones and the properties of their bases. This allows us to prove a characterization of reflexive cones in term of the absence of a subcone isomorphic to the positive cone of \ell_{1}. Moreover, the properties of some specific classes of reflexive cones are investigated. Namely, we consider the reflexive cones such that the intersection with the unit ball is norm compact, those ge…
