The variance conjecture on hyperplane projections of the lpn balls.

We show that for any 1 ≤ p ≤ ∞, the family of random vectors uniformly distributed on hyperplane projections of the unit ball of l_pⁿ verify the variance conjecture Var ∣X∣² ≤ C max ξ∈Sn^-1 E<X, ξ>² E∣X∣2^, where C depends on p but not on the dimension n or the hyperplane. We will also show a general result relating the variance conjecture for a random vector uniformly distributed on an isotropic convex body and the variance conjecture for a random vector uniformly distributed on any Steiner symmetrization of it. As a consequence we will have that the class of random vectors uniformly distributed on any Steiner symmetrization of an l_pⁿ -ball verify the variance conjecture. [ABSTRACT FROM AUTHOR]