Convex Hull algorithm - why it can't be computed using only comparisons
For a slightly more general way of looking at this, what you want to do is equivalent to saying you can reconstruct the convex hull from the orderings induced by projections onto the coordinate axes.  

To see you can't do this, just observe that for three points $a$, $b$, and $c$, knowing that $a &lt; c &lt; b$ in both projections doesn't let you figure out whether $c$ is above, below, or on the segment $ab$.  (Start with $c$ on the segment and then perturb it up or down by a small amount.)

This indicates that you'll need the "left turn" and "right turn" predicates.