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Real Analysis with Economic Applications - Chapter G

Chapter G Convexity One major reason why linear spaces are so important for geometric analysis is that they allow us to define the notion of “line segment” in algebraic terms. Among other things, this enables one to formulate, purely algebraically, the notion of “convex set” which figures majorly in a variety of branches of higher mathematics. | Chapter G Convexity One major reason why linear spaces are so important for geometric analysis is that they allow us to define the notion of line segment in algebraic terms. Among other things this enables one to formulate purely algebraically the notion of convex set which figures majorly in a variety of branches of higher mathematics.1 Immediately relevant for economic theory is the indispensable role played by convex sets in optimization theory. At least for economists this alone is enough of a motivation for taking on a comprehensive study of convex sets and related concepts. We begin the chapter with a fairly detailed discussion of convex sets and cones. Our emphasis is again on the infinite dimensional side of the picture. In particular we consider several examples that are couched within infinite dimensional linear spaces. After all one of our main objectives here is to provide some help for the novice to get over the sensation of shock that the strange behavior of infinite dimensional spaces may invoke at first. We also introduce the partially ordered linear spaces and discuss the important role played by convex cones thereof. Among the topics that are likely to be new to the reader are the algebraic interior and algebraic closure of subsets of a linear space. These notions are developed relatively leisurely for they are essential for the treatment of the high points of the chapter namely the fundamental extension and separation theorems in an arbitrary linear space. In particular we prove here the linear algebraic formulations of the Hahn-Banach Theorems on the extension of a linear functional and the separation of convex sets along with the Krein-Rutman Theorem on the extension of positive linear functionals. We then turn to Euclidean convex analysis and deduce Minkowski s Separating and Supporting Hyperplane Theorems these are among the most widely used theorems in economic theory as easy corollaries of our general results. As a final order of business .