To get dimension 4, I think the norm 
\begin{equation} \|a\| =  \max_{\{i,j\}}  |a_i-a_j| \vee \|a\|_\infty,
\end{equation}
where $a=(a_1,a_2,a_3,a_4)$, 
works. Again $X$ samples the unit vector basis uniformly.