Join Geometries

1. Front Matter

   Pages i-xxii

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2. The Join and Extension Operations in Euclidean Geometry

   • Walter Prenowitz, James Jantosciak

   Pages 1-45

3. The Abstract Theory of Join Operations

   • Walter Prenowitz, James Jantosciak

   Pages 46-123

4. The Generation of Convex Sets—Convex Hulls

   • Walter Prenowitz, James Jantosciak

   Pages 124-155

5. The Operation of Extension

   • Walter Prenowitz, James Jantosciak

   Pages 156-207

6. Join Geometries

   • Walter Prenowitz, James Jantosciak

   Pages 208-244

7. Linear Sets

   • Walter Prenowitz, James Jantosciak

   Pages 245-286

8. Extremal Structure of Convex Sets: Components and Faces

   • Walter Prenowitz, James Jantosciak

   Pages 287-341

9. Rays and Halfspaces

   • Walter Prenowitz, James Jantosciak

   Pages 342-367

10. Cones and Hypercones

    • Walter Prenowitz, James Jantosciak

    Pages 368-405

11. Factor Geometries and Congruence Relations

    • Walter Prenowitz, James Jantosciak

    Pages 406-436

12. Exchange Join Geometries—The Theory of Incidence and Dimension

    • Walter Prenowitz, James Jantosciak

    Pages 437-455

13. Ordered Join Geometries

    • Walter Prenowitz, James Jantosciak

    Pages 456-492

14. The Structure of Polytopes in an Ordered Geometry

    • Walter Prenowitz, James Jantosciak

    Pages 493-526

15. Back Matter

    Pages 527-535

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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.

• Equivalence
• Factor
• Finite
• Geometrie
• Konvexe Menge
• Maxima
• Morphism
• addition
• character
• function
• mathematics
• object
• presentation
• problem solving
• story

• Department of Mathematics, Brooklyn College, City University of New York, New York, USA

  Walter Prenowitz, James Jantosciak