Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 41, No. 2, pp. 437-454 (2000)

Asymptotics of Cross Sections for Convex Bodies

Ulrich Brehm, Jürgen Voigt

Fachrichtung Mathematik, Technische Universit{ä}t Dresden, D-01062 Dresden, Germany, e-mail:,

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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