The square negative correlation on l_p^n balls

TL;DR

This paper investigates the square negative correlation property of $ ext{l}_p^n$ balls, proving it holds for $p eq 2$ in certain cases and exploring projections onto specific hyperplanes.

Contribution

It establishes the square negative correlation property for $ ext{l}_p^n$ balls when $p eq 2$ and analyzes its behavior under specific orthogonal projections.

Findings

Property holds for $p eq 2$ with respect to any orthonormal basis.

The property is not always valid for $1 \\le p \\le 2$.

Projection onto hyperplanes orthogonal to the diagonal vector satisfies the property for all $p \\ge 1$ when $n$ is large.

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.