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Average volume of sections of star bodies. (English) Zbl 0973.52002
Milman, V. D. (ed.) et al., Geometric aspects of functional analysis. Proceedings of the Israel seminar (GAFA) 1996-2000. Berlin: Springer. Lect. Notes Math. 1745, 119-146 (2000).
Summary: We study the asymptotic behavior, as the dimension goes to infinity, of the volume of sections of the unit balls of the spaces $$\ell^n_q$$, $$0<q \leq\infty$$. We compute the precise asymptotics of the average volume of central sections and then prove a concentration inequality of exponential type. For the case of non-central hyperplane sections of the cube, we prove a local limit theorem confirming the conjecture on the asymptotically Gaussian dependence of the volume of sections on the distance from the hyperplane to the origin.
Note that a weak limit theorem was established very recently in [M. Anttilla, K. Ball and I. Perissinaki, ‘The central limit problem for convex bodies’, preprint] for a larger class of bodies. Our calculations are based on connections between volume and the Fourier transform.
For the entire collection see [Zbl 0949.00025].

##### MSC:
 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces)