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Asymptotics of cross sections for convex bodies. (English) Zbl 0983.52004
A normed isotropic convex body $$K\subset \mathbb{R}^n$$ is a convex compact set of volume 1 whose ellipsoid of inertia is a ball centred at the origin. For a unit vector $$u$$ and $$t\in\mathbb{R}$$, let $$\varphi_{K,u}(t)$$ denote the $$(n-1)$$-dimensional volume of the intersection of $$K$$ with the hyperplane $$\{x\cdot u = t\}$$.
The authors study a ‘central limit property’ for normed isotropic $$K$$, which says that $$\varphi_{K,u}$$ is, for large $$n$$ and most $$u$$, close to a Gaussian density. Results of this central limit type are shown for cubes, balls, cross polytopes and regular simplices, whereby closeness of the densities is expressed in terms of the $$L_1$$- and $$L_\infty$$-norms.
Some details of the lengthy calculations are omitted, they are available in the electronic version of the paper.

##### MSC:
 52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry) 60F25 $$L^p$$-limit theorems
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