The variance conjecture on hyperplane projections of the ℓnp balls

• David Alonso ^[1] ; Jesús Bastero ^[1]
  1. [1] Universidad de Zaragoza

     Universidad de Zaragoza

     Zaragoza, España

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 34, Nº 2, 2018, págs. 879-904
• Idioma: inglés
• DOI: 10.4171/RMI/1007
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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.