# A Berry-Esseen type inequality for convex bodies with an unconditional basis

@article{Klartag2007ABT,
title={A Berry-Esseen type inequality for convex bodies with an unconditional basis},
author={Bo'az Klartag},
journal={Probability Theory and Related Fields},
year={2007},
volume={145},
pages={1-33}
}
• B. Klartag
• Published 7 May 2007
• Mathematics
• Probability Theory and Related Fields
Suppose X = (X1, . . . , Xn) is a random vector, distributed uniformly in a convex body $${K \subset \mathbb R^n}$$ . We assume the normalization $${\mathbb E X_i^2 = 1}$$ for i = 1, . . . , n. The body K is further required to be invariant under coordinate reflections, that is, we assume that (±X1, . . . , ±Xn) has the same distribution as (X1, . . . , Xn) for any choice of signs. Then, we show that$$\mathbb E \left( \, |X| - \sqrt{n} \, \right)^2 \leq C^2,$$where C  ≤  4 is a positive…

## Figures from this paper

Poincaré inequality in mean value for Gaussian polytopes
AbstractLet KN = [±G1, . . . , ±GN] be the absolute convex hull of N independent standard Gaussian random points in $${\mathbb R^n}$$ with N ≥ n. We prove that, for any 1-Lipschitz function
The variance conjecture on hyperplane projections of l_p^n balls
• Mathematics
• 2016
We show that for any $1\leq p\leq\infty$, the family of random vectors uniformly distributed on hyperplane projections of the unit ball of $\ell_p^n$ verify the variance conjecture $$Berry-Esseen Bounds for Projections of Coordinate Symmetric Random Vectors • Mathematics • 2009 For a coordinate symmetric random vector (Y_1,\ldots,Y_n)={\bf Y} \in \mathbb{R}^n, that is, one satisfying (Y_1,\ldots,Y_n)=_d(e_1Y_1,\ldots,e_nY_n) for all (e_1,\ldots,e_n) \in \{-1,1\}^n, More on logarithmic sums of convex bodies We prove that the log-Brunn-Minkowski inequality (log-BMI) for the Lebesque measure in dimension n would imply the log-BMI and, therefore, the B-conjecture for any log-concave density in dimension Spectral gap for some invariant log-concave probability measures We show that the conjecture of Kannan, Lov\'{a}sz, and Simonovits on isoperimetric properties of convex bodies and log-concave measures, is true for log-concave measures of the form \rho(|x|_B)dx The KLS isoperimetric conjecture for generalized Orlicz balls • Mathematics The Annals of Probability • 2018 What is the optimal way to cut a convex bounded domain K in Euclidean space (\mathbb{R}^n,|\cdot|) into two halves of equal volume, so that the interface between the two halves has least surface Gaussian fluctuations for high-dimensional random projections of \ell_{p}^{n}-balls • Mathematics Bernoulli • 2019 In this paper, we study high-dimensional random projections of \ell_p^n-balls. More precisely, for any n\in\mathbb N let E_n be a random subspace of dimension k_n\in\{1,\ldots,n\} and X_n Some remarks about the maximal perimeter of convex sets with respect to probability measures In this note we study the maximal perimeter of a convex set in \mathbb{R}^n with respect to various classes of measures. Firstly, we show that for a probability measure \mu on  \mathbb{R}^n, Limit theorems for quadratic variations of the Lei–Nualart process • Physics • 2018 Let X be a Lei–Nualart process with Hurst index $$H\in (0, 1)$$, $$Z_{1}$$ be an Hermite random variable. For any $$n \ge 1$$, set$$V_{n}=\sum _{k=0}^{n-1}\left[ n^{2H}(\varDelta _k
Local Lp-Brunn-Minkowski inequalities for p<1
• Mathematics
• 2017
The $L^p$-Brunn-Minkowski theory for $p\geq 1$, proposed by Firey and developed by Lutwak in the 90's, replaces the Minkowski addition of convex sets by its $L^p$ counterpart, in which the support

## References

SHOWING 1-10 OF 62 REFERENCES
A central limit theorem for convex sets
We show that there exists a sequence $\varepsilon_n\searrow0$ for which the following holds: Let K⊂ℝn be a compact, convex set with a non-empty interior. Let X be a random vector that is distributed
On convex perturbations with a bounded isotropic constant
Abstract.Let $$K \subset {\user2{\mathbb{R}}}^{n}$$ be a convex body and ɛ  > 0. We prove the existence of another convex body $$K' \subset {\user2{\mathbb{R}}}^{n}$$, whose Banach–Mazur distance
Moment inequalities and central limit properties of isotropic convex bodies
• Mathematics
• 2002
Abstract. The object of our investigations are isotropic convex bodies $K\subseteq \mathbb{R}^n$, centred at the origin and normed to volume one, in arbitrary dimensions. We show that a certain
On the Volume of the Intersection of Two L n p Balls
• Mathematics
• 1989
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
Concentration of mass on convex bodies
Abstract.We establish sharp concentration of mass inequality for isotropic convex bodies: there exists an absolute constant c >  0 such that if K is an isotropic convex body in $$\mathbb{R}^{n}$$,
Inequalities between dirichlet and Neumann eigenvalues
• Physics
• 1986
The purpose of this paper is to derive some inequalities of the form $${u _k} + R < {ambda _k} for k = 1, 2, ...$$ (1.1) between the eigenvalues λ 1 < λ 2 ≦ ... of the Dirichlet problem
Uniform almost sub-Gaussian estimates for linear functionals on convex sets
A well-known consequence of the Brunn-Minkowski inequality, is that the distribution of a linear functional on a convex set has a sub-exponential tail. That is, for any dimension n, a convex set K ⊂
The Square Negative Correlation Property for Generalized Orlicz Balls
Antilla, Ball and Perissinaki proved that the squares of coordinate functions in $\ell_p^n$ are negatively correlated. This paper extends their results to balls in generalized Orlicz norms on R^n.
On the role of convexity in isoperimetry, spectral gap and concentration
We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a-priori
On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint
• Mathematics
• 1999
among all competitors F C Q such that \F\ = \E\ and such that ||XF — X^IILUQ) < ^ f some 6 > 0. We prove that when Q is convex, the boundary dE Pi O is connected, or else dE Pi O consists of parallel