The square negative correlation property on $\ell_p^n$- balls

Abstract

In this paper we prove that for any p ∈ [2,∞) 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 ≤ p ≤ 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, …, 1). In this case, the property holds for all p ≥ 1 and n large enough.