  We define a new cutting plane closure for pure integer programs called the
two-halfspace closure. It is a natural generalization of the well-known
Chv\'atal-Gomory closure. We prove that the two-halfspace closure is
polyhedral. We also study the corresponding $2$-halfpsace rank of any valid
inequality and show that it is at most the split rank of the inequality.
Moreover, while the split rank can be strictly larger than the two-halfspace
rank, the split rank is at most twice the two-halfspace rank. A key step of our
analysis shows that the split closure of a rational polyhedron can be obtained
by considering the split closures of all $k$-dimensional (rational) projections
of the polyhedron, for any fixed $k\geq 2$. This result may be of independent
interest.
