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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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##### References:
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