Asymptotics of Cross Sections for Convex Bodies

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 xed 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. MSC 2000: 52A21 (primary), 60F25 (secondary)