Here is a similar but more difficult problem than the one I asked already:

Consider two points $p_1$ and $p_2$ in the euclidean plane and a set of $n$ concentric circles around $p_1$ and a set of $m$ concentric circles around $p_2$, what is the maximum number of points in convex position on the grid made by intersection of these two families of concentric circles?

Remark: (with condition) what if $p_1$ and $p_2$ be two of these convex points and other points said to lie on one side of the line $p_1 p_2$ ? (I mean when $p_1$, $p_2$ and $N-2$ other points (which are from the grid) form a convex $N$-gon, how large this $N$ can be in terms of n and m ? (my own guess: $N$ is of order $m+n$, $N$ can not be multiplicative in $n$ and $m$)

(i think this problem is solvable but hard , and absolutely worth thinking on .thank you )