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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
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