On convex polygons in cartesian products

We study several problems concerning convex polygons whose vertices lie on a grid defined by the Cartesian product of two sets of n real numbers, using each coordinate at most once. First, we prove that all such grids contain a convex polygon with Ω(log n) vertices and that this bound is asymptotically tight. Second, we present two polynomial-time algorithms that find the largest
convex polygon of a restricted type. These algorithms give an approximation of the unrestricted case. It is unknown whether the unrestricted problem can be solved in polynomial time.