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The central limit problem for convex bodies. (English) Zbl 1033.52003
A symmetric convex body \(K \subset R^d\), of volume 1, is in isotropic position if \(\int \langle x,\theta \rangle ^2dx=\rho^2\) for all unit vectors \(\theta \in R^d\). Then the random variable \(X_{\theta}:x\to \langle x,\theta \rangle\) has expectation 0 and variance \(\rho^2\).
The main result is that if \(K\) satisfies a certain concentration hypothesis, then for all \(t>0\) and all unit vectors \(\theta\), except for a set of small spherical measure, the probability \(P(| X_{\theta}| >t)\) is very close to \(P(| \gamma| >t)\) where \(\gamma\) is a Gaussian random variable with mean 0 and variance \(\rho^2\). The concentration hypothesis is shown to hold for two large classes of convex bodies; both classes contain all \(\ell^p_n\) balls.

MSC:
52A22 Random convex sets and integral geometry (aspects of convex geometry)
60F05 Central limit and other weak theorems
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