TITLE: "Positive systems" in n * the (n-1)-simplex
QUESTION [11 upvotes]: Let S := the nonnegative integer solutions to {$a_1 + ... + a_n = n$},
and center := (1,1,1,...,1).
Call a vector v generic if v.s = v.center <=> s = center.
Then each generic v defines a positive system in S,
the subset { s in S : v.s > v.center }.
Already at n=3 it is possible for one positive system to contain another.
Up to permutation, we may as well take v strictly decreasing.
Having done so, is there a reasonable way to classify 
the maximal positive systems?

REPLY [2 votes]: If you replace S by the lattice points in the positive orthant, and forget the condition that the hyperplane passes through a particular point, and require that the hyperplane only cuts off finitely many points, then you are looking at things called "corner cuts" by Onn and Sturmfels in Cutting corners.
Anyway, I'm not sure if this will help, but maybe it'll suggest something.