Join Geometries: A Theory of Convex Sets and Linear Geometry (Undergraduate Texts in Mathematics)

The main object of this book is to reorient and revitalize classical geometry in a way that will bring it closer to the mainstream of contemporary mathematics. The postulational basis of the subject will be radically revised in order to construct a broad-scale and conceptually unified treatment. The familiar figures of classical geometry-points, segments, lines, planes, triangles, circles, and so on-stem from problems in the physical world and seem to be conceptually unrelated. However, a natural setting for their study is provided by the concept of convex set, which is compara­ tively new in the history of geometrical ideas. The familiarfigures can then appear as convex sets, boundaries of convex sets, or finite unions of convex sets. Moreover, two basic types of figure in linear geometry are special cases of convex set: linear space (point, line, and plane) and halfspace (ray, halfplane, and halfspace). Therefore we choose convex set to be the central type of figure in our treatment of geometry. How can the wealth of geometric knowledge be organized around this idea? By defini­ tion, a set is convex if it contains the segment joining each pair of its points; that is, if it is closed under the operation of joining two points to form a segment. But this is precisely the basic operation in Euclid. Front Matter....Pages i-xxii The Join and Extension Operations in Euclidean Geometry....Pages 1-45 The Abstract Theory of Join Operations....Pages 46-123 The Generation of Convex Sets—Convex Hulls....Pages 124-155 The Operation of Extension....Pages 156-207 Join Geometries....Pages 208-244 Linear Sets....Pages 245-286 Extremal Structure of Convex Sets: Components and Faces....Pages 287-341 Rays and Halfspaces....Pages 342-367 Cones and Hypercones....Pages 368-405 Factor Geometries and Congruence Relations....Pages 406-436 Exchange Join Geometries—The Theory of Incidence and Dimension....Pages 437-455 Ordered Join Geometries....Pages 456-492 The Structure of Polytopes in an Ordered Geometry....Pages 493-526 Back Matter....Pages 527-535