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A probabilistic approach to the geometry of the $$\ell^n_p$$-ball. (English) Zbl 1071.60010
Authors’ abstract: This article investigates, by probabilistic methods, various geometric questions on $$B^n_{p}$$, the unit ball of $$l^n_{p}$$. We propose realizations in terms of independent random variables of several distributions on $$B^n_{p}$$, including the normalized volume measure. These representations allow us to unify and extend the known results of the sub-independence of coordinates slabs in $$B^{n}_{p}$$. As another application, we compute moments of linear functionals on $$B^{n}_{p}$$, which gives sharp constants in Khinchin’s inequalities on $$B^n_{p}$$ and determines the $$\psi_2$$-constant of all directions on $$B^n_p$$. We also study the extremal values of several Gaussian averages on sections of $$B^n_{p}$$ (including mean width and $$l$$-norm), and derive several monotonicity results as $$p$$ varies. Applications to balancing vectors in $$l_2$$ and to covering numbers of polyhedra complete the exposition.

##### MSC:
 60E15 Inequalities; stochastic orderings 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces) 52A38 Length, area, volume and convex sets (aspects of convex geometry) 52A40 Inequalities and extremum problems involving convexity in convex geometry
##### Keywords:
Gaussian measure; extremal sections
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##### References:
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