  We show that any random vector uniformly distributed on any hyperplane
projection of $B_1^n$ or $B_\infty^n$ verifies the variance conjecture
$$\text{Var}|X|^2\leq C\sup_{\xi\in S^{n-1}}\E<X,\xi>^2\E|X|^2.$$ 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.
