# A central limit theorem for convex sets

@article{Klartag2007ACL,
title={A central limit theorem for convex sets},
author={Bo'az Klartag},
journal={Inventiones mathematicae},
year={2007},
volume={168},
pages={91-131}
}
• B. Klartag
• Published 29 April 2006
• Mathematics
• Inventiones mathematicae
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 uniformly in K. Then there exist a unit vector θ in ℝn, t0∈ℝ and σ>0 such that $$\sup_{A\subset\mathbb{R}}\left|\textit{Prob}\,\{\langle X,\theta\rangle\in A\}-\frac{1}{\sqrt{2\pi\sigma}}\int_Ae^{-\frac{(t - t_0)^2}{2\sigma^2}} dt\right|\leq\varepsilon_n,\qquad{(\ast)}$$ where the supremum runs over…
221 Citations

• Mathematics
• 2014

### 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

### 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 ⊂

### Sudakov's typical marginals, random linear functionals and a conditional central limit theorem

V.N. Sudakov [Sud78] proved that the one-dimensional marginals of a high-dimensional second order measure are close to each other in most directions. Extending this and a related result in the

### 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.

### Asymptotic behavior of averages of k-dimensional marginals of measures on Rn

• Mathematics
• 2005
We study the asymptotic behaviour, as n → ∞, of the Lebesgue measure of the set {x ∈ K ; |PE(x)| ≤ t} for a random kdimensional subspace E ⊂ Rn and K ⊂ Rn in a certain class of isotropic bodies. For

### Concentration of mass and central limit properties of isotropic convex bodies

We discuss the following question: Do there exist an absolute constant c > 0 and a sequence Φ(n) tending to infinity with n, such that for every isotropic convex body K in R n and every t ≥ 1 the

### On logarithmic concave measures and functions

The purpose of the present paper is to give a new proof for the main theorem proved in [3] and develop further properties of logarithmic concave measures and functions. Having in mind the

### The geometry of logconcave functions and sampling algorithms

• Computer Science, Mathematics
Random Struct. Algorithms
• 2007
These results are applied to analyze two efficient algorithms for sampling from a logconcave distribution in n dimensions, with no assumptions on the local smoothness of the density function.