  We study several problems concerning convex polygons whose vertices lie in a
Cartesian product of two sets of $n$ real numbers (for short, \emph{grid}).
First, we prove that every such grid contains $\Omega(\log n)$ points in convex
position and that this bound is tight up to a constant factor. We generalize
this result to $d$ dimensions (for a fixed $d\in \mathbb{N}$), and obtain a
tight lower bound of $\Omega(\log^{d-1}n)$ for the maximum number of points in
convex position in a $d$-dimensional grid. Second, we present polynomial-time
algorithms for computing the longest $x$- or $y$-monotone convex polygonal
chain in a grid that contains no two points with the same $x$- or
$y$-coordinate. We show that the maximum size of a convex polygon with such
unique coordinates can be efficiently approximated up to a factor of $2$.
Finally, we present exponential bounds on the maximum number of point sets in
convex position in such grids, and for some restricted variants. These bounds
are tight up to polynomial factors.
