On the Complexity of Polytopes in LI(2)

In this paper we consider polytopes given by systems of n inequalities in d variables, where every inequality has at most two variables with nonzero coefficient. We denote this family by LI(2). We show that despite of the easy algebraic structure, polytopes in LI(2) can have high complexity. We construct a polytope in LI(2), whose number of vertices is almost the number of vertices of the dual cyclic polytope, the difference is a multiplicative factor of depending on d and in particular independent of n. Moreover we show that the dual cyclic polytope can not be realized in LI(2).