The variance conjecture on some polytopes

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