# Intersection Cuts with Infinite Split Rank

## TL;DR

This paper characterizes when intersection cuts in mixed integer linear programs have finite split rank, extending previous results to multiple integer variables through a geometric property of boundary points.

## Contribution

It introduces a general condition, the '2-hyperplane property', that determines the finiteness of split rank for intersection cuts with any number of integer variables.

## Key findings

- Split rank is finite iff boundary integer points satisfy the 2-hyperplane property.
- Extends Dey-Louveaux characterization from two to multiple integer variables.
- Provides a geometric criterion for analyzing intersection cuts.

## Abstract

We consider mixed integer linear programs where free integer variables are expressed in terms of nonnegative continuous variables. When this model only has two integer variables, Dey and Louveaux characterized the intersection cuts that have infinite split rank. We show that, for any number of integer variables, the split rank of an intersection cut generated from a bounded convex set $P$ is finite if and only if the integer points on the boundary of $P$ satisfy a certain "2-hyperplane property". The Dey-Louveaux characterization is a consequence of this more general result.