The variance conjecture on some polytopes

Abstract

We show that any random vector uniformly distributed on any hyperplane projection of $B_1^n$ or $B_{\infty }^n$ verifies the variance conjecture $$\text{Var}|X|^2\leq C\sup_{ξ\in S^{n-1}}\E<X,ξ>^2\E|X|^2.$$ Furthermore, a random vector uniformly distributed on a hyperplane projection of $B_{\infty }^n$ verifies a negative square correlation property and consequently any of its linear images verifies the variance conjecture.