## Abstract

Let S ℤ^{n} satisfy the property that conv(S) n ℤ^{n} = S. Then a convex set K is called an S-free convex set if int(K) S = Ø. A maximal S-free convex set is an S-free convex set that is not properly contained in any S-free convex set. We show that maximal S-free convex sets are polyhedra. This result generalizes a result of Basu et al. [SIAM J. Discrete Math., 24 (2010), pp. 158- 168] for the case where S is the set of integer points in a rational polyhedron and a result of Lovász [Mathematical Programming: Recent Developments and Applications, M. Iri and K. Tanabe, eds., Kluwer, Dordrecht, 1989, pp. 177-210] and Basu et al. [Math. Oper. Res., 35 (2010), pp. 704-720] for the case where S is the set of integer points in some affine subspace of Rn.

Original language | English |
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Pages (from-to) | 379-393 |

Number of pages | 15 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 25 |

Issue number | 1 |

DOIs | |

State | Published - 2011 |

Externally published | Yes |

## Keywords

- Cutting planes
- Integer nonlinear programming
- Maximal lattice-free convex sets