In this chapter, we develop the basic results of the theory of convex polyhedra. This is a large area of research that has been studied from many different points of view. Within optimization, it is very important in linear programming, especially in connection with the simplex method for solving linear programs. The choice of the topics we treat in this chapter is dictated mostly by the needs of optimization. However, we do not have space to treat the extensive body of work concerning the combinatorial theory of convex polyhedra, some of which is intimately related to the simplex method and its variants. The interested reader may consult the books [115, 50, 274] for more information on this topic and the book  for differential-geometric questions regarding convex polyhedra.
KeywordsVariational Inequality Linear Inequality Convex Polyhedron Polyhedral Cone Polar Cone
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