# 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} }

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

## 90 Citations

Poincaré inequality in mean value for Gaussian polytopes

- Mathematics
- 2012

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

- Mathematics
- 2014

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

- Mathematics
- 2010

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

- MathematicsThe 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

- MathematicsBernoulli
- 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

- Mathematics
- 2019

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

- Mathematics
- 2007

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

- Mathematics
- 2006

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

- Mathematics
- 2006

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

- Mathematics
- 2007

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

- Mathematics
- 2007

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

- Mathematics
- 2007

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…