Order Homomorphisms in Euclidean Space

Abstract

In this chapter we discuss the construction of order homomorphisms on closed convex subsets of N-dimensional euclidean space. Although the techniques used below only apply in the context of euclidean space (since they make essential and intuitively appealing use of the euclidean distance function), they have two substantial advantages over many of the more widely applicable methods described in later chapters: first, they enable us to write down the order homomorphism we seek; and secondly (as we shall see in Chapter 8), they yield order homomorphisms that are upper semicontin-uous, or even continuous, with respect to an appropriate topology on the set of preference relations (in other words, they yield representations with properties of continuity relative to both preferences and commodities).