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… 

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

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

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

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

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 $X_n$

Limit theorems for quadratic variations of the Lei–Nualart process

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

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

Interpolating Thin-Shell and Sharp Large-Deviation Estimates for Lsotropic Log-Concave Measures

Given an isotropic random vector X with log-concave density in Euclidean space $${\mathbb{R}^n}$$ , we study the concentration properties of |X| on all scales, both above and below its expectation.
...

References

SHOWING 1-10 OF 61 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

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

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

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

The “hot spots” conjecture for domains with two axes of symmetry

Consider a convex planar domain with two axes of symmetry. We show that the maximum and minimum of a Neumann eigenfunction with lowest nonzero eigenvalue occur at points on the boundary only. We
...