The Variance Conjecture on Some Polytopes
The Variance Conjecture on Some Polytopes
We show that any random vector uniformly distributed on any hyperplane projection of B 1 n or B ∞ n verifies the variance conjecture $$\text{Var }\vert X{\vert }^{2} \leq C\sup\limits_{ \xi \in {S}^{n-1}}\mathbb{E}\langle X,{\xi \rangle }^{2}\mathbb{E}\vert X{\vert }^{2}.$$ Furthermore, a random vector uniformly distributed on a hyperplane projection of B …