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Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 41, No. 2, pp. 437-454 (2000)
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Asymptotics of Cross Sections for Convex Bodies

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Ulrich Brehm, Jürgen Voigt

Fachrichtung Mathematik, Technische Universit{ä}t Dresden, D-01062 Dresden, Germany, e-mail: brehm@math.tu-dresden.de, voigt@math.tu-dresden.de

**Abstract:** For normed isotropic convex bodies in $\R^n$ we investigate the behaviour of the $(n-1)$-dimensional volume of intersections with hyperplanes orthogonal to a fixed direction, considered as a function of the distance of the hyperplane to the origin. It is a conjecture that for arbitrary normed isotropic convex bodies and random directions this function - with high probability - is close to a Gaussian density, for large dimension $n$. This would be a kind of central limit theorem. We determine this function explicitly for several families of convex bodies and several directions and obtain results concerning the asymptotic behaviour supporting the conjecture.

**Keywords:** convex body, isotropic, cross section, central limit theorem, marginal distribution

**Classification (MSC2000):** 52A21; 60F25

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