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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.

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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