A note on the split rank of intersection cuts

Abstract

In this note, we present a simple geometric argument to determine a lower bound on the split rank of intersection cuts. As a first step of this argument, a polyhedral subset of the lattice-free convex set that is used to generate the intersection cut is constructed. We call this subset the restricted lattice-free set. It is then shown that $${\lceil \log_2 (l)\rceil}$$ is a lower bound on the split rank of the intersection cut, where l is the number of integer points lying on the boundary of the restricted lattice-free set satisfying the condition that no two points lie on the same facet of the restricted lattice-free set. The use of this result is illustrated by obtaining a lower bound of $${\lceil \log_2( n+1) \rceil}$$ on the split rank of n-row mixing inequalities.