Dantzig's development of linear programming into one of the most
applicable optimization techniques has spread interest in the algebra of
linear inequalities, the geometry of polyhedra, the topology of convex
sets, and the analysis of convex functions. It is the goal of this
volume to provide a synopsis of these topics, and thereby the
theoretical back- ground for the arithmetic of convex optimization to be
treated in a sub- sequent volume. The exposition of each chapter is
essentially independent, and attempts to reflect a specific style of
mathematical reasoning. The emphasis lies on linear and convex duality
theory, as initiated by Gale, Kuhn and Tucker, Fenchel, and v. Neumann,
because it represents the theoretical development whose impact on modern
optimi- zation techniques has been the most pronounced. Chapters 5 and 6
are devoted to two characteristic aspects of duality theory: conjugate
functions or polarity on the one hand, and saddle points on the other.
The Farkas lemma on linear inequalities and its generalizations,
Motzkin's description of polyhedra, Minkowski's supporting plane theorem
are indispensable elementary tools which are contained in chapters 1, 2
and 3, respectively. The treatment of extremal properties of polyhedra
as well as of general convex sets is based on the far reaching work of
Klee. Chapter 2 terminates with a description of Gale diagrams, a
recently developed successful technique for exploring polyhedral
structures.