Lattice convex $n$-gon $(n=4k+b)$ with minimum $L_\infty$ diameter $MD(n)$ can be constructed as the Minkowski sum of a centrally symmetric lattice convex $4k$-gon and a non-symmetric part, so called Basic $b$-tuple. This paper investigates the conditions by which a family of Basic $b$-tuples can and cannot be used to build optimum lattice convex $n$-gons for large classes of $n$. Solutions for five special small values of $n$ are presented. It has been formerly shown that seventeen suitably chosen families of Basic $b$-tuples with $b\leq11$ can cover all the remaining values.