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Gaussian behaviour and average of marginals for convex bodies. (English) Zbl 1163.46006

The first part of the paper is a short survey of recent results on the central limit theorem for convex bodies. In the second part, the authors provide a sketch of a proof of a central limit type theorem for a random \(k\)-dimensional subspace of \(\mathbb{R}^n\) and an isotropy body with bounded isotropy constant. This theorem extends a previous 1-dimensional result of S.Sodin [Lect.Notes in Math.1910, 271–295 (2007; Zbl 1142.60016)]. The complete proof as well as some related results appear in J.Bastero and J.Bernués [Stud.Math.190, No.1, 1–31 (2009; Zbl 1157.60011)].

MSC:

46B07 Local theory of Banach spaces
46B09 Probabilistic methods in Banach space theory
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
60F05 Central limit and other weak theorems
60D05 Geometric probability and stochastic geometry
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
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