A probabilistic approach to the geometry of the lnp-ball

This article investigates, by probabilistic methods, various geometric questions on B-p(n), the unit ball of l(p)(n). We propose realizations in terms of independent random variables of several distributions on B-p(n), including the P normalized volume measure. These representations allow us to unify and extend the known results of the sub-independence of coordinate slabs in B-p(n), As another application, we compute moments of linear functionals on B-p(n) which gives sharp constants in Khinchine's inequalities on B-p(n) and determines the psi(2)-constant of all directions on B-p(n). We also study the extremal values of several Gaussian averages on sections of B-p(n) (including mean width and l-norm), and derive several monotonicity results as varies. Applications to balancing vectors in l(2) and to covering numbers of polyhedra complete the exposition.