The square negative correlation on l_p^n balls

Abstract

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.