Bound the number of vertices on the convex hull of a self cartesian product

Question

Given a finite set $A \subset \mathbb{R}$ with $n$ elements. One can construct a set $B$ in $\mathbb{R}^2$ using the cartesian product: $B = A \times A \subset \mathbb{R}^2$.

Is there any tight bound of the number of vertices on the convex hull of $B$?