Resumen de The variance conjecture on hyperplane projections of the ℓnp balls

David Alonso Gutiérrez , Jesús Miguel Bastero Eleizalde

• We show that for any 1≤p≤∞, the family of random vectors uniformly distributed on hyperplane projections of the unit ball of ℓnp verify the variance conjecture Var|X|2≤Cmaxξ∈Sn−1E⟨X,ξ⟩2E|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 ℓnp-ball verify the variance conjecture.