  We show that for any $1\leq p\leq\infty$, the family of random vectors
uniformly distributed on hyperplane projections of the unit ball of $\ell_p^n$
verify the variance conjecture $$ \textrm{Var}\,|X|^2\leq C\max_{\xi\in
S^{n-1}}\mathbb{E}\langle X,\xi\rangle^2\mathbb{E}|X|^2, $$ 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 $\ell_p^n$-ball verify the variance
conjecture.
