I have a set of lattice points S in R^n (listed in memory in a computer for n=8 say). I want to computationally certify that they do *not* form the lattice points of a convex polytope P in R^n. (Ex. S={-1,1} in R^1.) Is there an easy (and hopefully efficient enough) way to do this?

Remarks:

I don't have much feel for the (many) points themselves at the moment. For instance, I don't know what are vertices of the convex hull of S (or how to find that).

Assuming I could find a description of the convex hull, I get scared about a solution that says "now figure out what the lattice points are and compare with S", because of efficiency concerns.

Thank you!