  In this work we study $d$-dimensional majorant properties. We prove that a
set of frequencies in ${\mathbb Z}^d$ satisfies the strict majorant property on
$L^p([0,1]^d)$ for all $p> 0$ if and only if the set is affinely independent.
We further construct three types of violations of the strict majorant property.
Any set of at least $d+2$ frequencies in ${\mathbb Z}^d$ violates the strict
majorant property on $L^p([0,1]^d)$ for an open interval of $p \not\in 2
{\mathbb N}$ of length 2. Any infinite set of frequencies in ${\mathbb Z}^d$
violates the strict majorant property on $L^p([0,1]^d)$ for an infinite
sequence of open intervals of $p \not\in 2 {\mathbb N}$ of length $2$. Finally,
given any $p>0$ with $p \not\in 2{\mathbb N}$, we exhibit a set of $d+2$
frequencies on the moment curve in ${\mathbb R}^d$ that violate the strict
majorant property on $L^p([0,1]^d).$
