ABSTRACT
This chapter explores relationships among the duality theorems. It also gives several new results involving separation, convexity, complementary slackness, and two new concepts, facets and polarity. The law of conservation of difficulty is a rule of thumb. It says that if a mathematical result is deep or difficult, it can't be proved easily unless another deep result is employed. There are a bunch of theorems in linear programming which all take some work to prove from scratch. The chapter discusses some of the relatively easy paths between pairs of the theorems. It introduces polarity, a fascinating duality between pairs of polyhedra wherein the vertices and facets exchange roles. Polarity forms pairs of polyhedra each of whose extreme points are the facets of the other. Gomory, Johnson, and Evans performed similar shooting experiments on master cyclic group polyhedra, though they shot from outside the polyhedron.
