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Corner Polyhedra and Maximal Lattice-free Convex Sets : A Geometric Approach to Cutting Plane Theory

Corner Polyhedra were introduced by Gomory in the early 60s and were studied by Gomory and Johnson. The importance of the corner polyhedron is underscored by the fact that almost all “generic” cutting planes, both in the theoretical literature as well as ones used in practice, are valid for the corner polyhedron. So the corner polyhedron can be viewed as a unifying structure from which many of the known cutting planes can be derived. Moreover, the corner polyhedron has recently attracted a lot of attention as a potential source of new kinds of cutting planes. It has also been observed that valid inequalities for the corner polyhedron have an intimate connection with maximal lattice-free convex sets in R. This connection provides a geometric way of understanding and analyzing these valid inequalities. Such an approach often yields new insights into properties of existing cutting planes, as well as provides a novel way of analyzing new families of cutting planes. The goal of this thesis is to explore this interplay of maximal lattice-free convex bodies and cutting plane theory. We also hope to obtain new insights into the structure of the corner polyhedron with the help of this approach.