Number of convex polygons in chess board

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how many convex k-sided polygons are made of intersection points of a $m\times n$ chess board? i tried to determine the properties of a convex k_sided polygon and i got below statement : "in convex k_sided polygons there is no side with line equation of $y=f(x)$ that has answer with line equations of exactly two other sides" (i tried this for five-sided polygons and it was correct)