Cones and Hypercones
This chapter builds on the preceding chapter, and like its predecessor consists of two well-defined parts—a study of cones followed by a parallel study of their generalization, hypercones. A cone is defined to be a union of a family of rays that have a common endpoint. Cones which are convex, and of these, especially those which are “pointed”, are of most importance. The problem of how convex cones are generated by their rays receives a major share of our attention. Naturally the case of the polyhedral cone—a convex cone generated by a finite family of rays—is considered. The notion of an extreme ray of a convex cone, as an analogue of an extreme point of an arbitrary convex set, is clarified and examined closely. Then the study of cones concludes with the determination of conditions for a polyhedral cone to be generated by its extreme rays.
KeywordsExtreme Point Convex Cone Euclidean Geometry Finite Family Polyhedral Cone
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