Cook, Kannan and Schrijver's example revisited

In 1990, Cook, Kannan and Schrijver proved that the split closure (the 1st 1-branch split closure) of a polyhedron is again a polyhedron. They also gave an example of a 3-dimensional mixed-integer polytope whose 1-branch split rank is infinite. We generalize this example to a family of high-dimensional polytopes and present a closed-form description of the kth 1-branch split closure of these polytopes for any k ≥ 1. Despite the fact that the m-branch split rank of the (m + 1)dimensional polytope in this family is 1, we show that the 2-branch split rank of the (m+1)-dimensional polytope is infinite when m ≥ 3. We discuss whether the (m − 1)-branch split rank of the (m + 1)dimensional polytope of the family is infinite for any m ≥ 2. Join work with Jean-Philippe Richard.