  In this paper we prove that for any $p\in[2,\infty)$ the $\ell_p^n$ unit
ball, $B_p^n$, satisfies the square negative correlation property with respect
to every orthonormal basis, while we show it is not always the case for $1\le
p\le 2$. In order to do that we regard $B_p^n$ as the orthogonal projection of
$B_p^{n+1}$ onto the hyperplane $e_{n+1}^\perp$. We will also study the
orthogonal projection of $B_p^n$ onto the hyperplane orthogonal to the diagonal
vector $(1,\dots,1)$. In this case, the property holds for all $p\ge 1$ and $n$
large enough.
