# Join Geometries

## A Theory of Convex Sets and Linear Geometry

Textbook

Part of the Undergraduate Texts in Mathematics book series (UTM)

1. Front Matter
Pages i-xxii
2. Walter Prenowitz, James Jantosciak
Pages 1-45
3. Walter Prenowitz, James Jantosciak
Pages 46-123
4. Walter Prenowitz, James Jantosciak
Pages 124-155
5. Walter Prenowitz, James Jantosciak
Pages 156-207
6. Walter Prenowitz, James Jantosciak
Pages 208-244
7. Walter Prenowitz, James Jantosciak
Pages 245-286
8. Walter Prenowitz, James Jantosciak
Pages 287-341
9. Walter Prenowitz, James Jantosciak
Pages 342-367
10. Walter Prenowitz, James Jantosciak
Pages 368-405
11. Walter Prenowitz, James Jantosciak
Pages 406-436
12. Walter Prenowitz, James Jantosciak
Pages 437-455
13. Walter Prenowitz, James Jantosciak
Pages 456-492
14. Walter Prenowitz, James Jantosciak
Pages 493-526
15. Back Matter
Pages 527-535

### Introduction

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.

### Keywords

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

#### Authors and affiliations

1. 1.Department of MathematicsBrooklyn College, City University of New YorkNew YorkUSA

### Bibliographic information

• Book Title Join Geometries
• Book Subtitle A Theory of Convex Sets and Linear Geometry
• Authors W. Prenowitz
J. Jantosciak
• Series Title Undergraduate Texts in Mathematics
• DOI https://doi.org/10.1007/978-1-4613-9438-9
• Copyright Information Springer-Verlag New York 1979
• Publisher Name Springer, New York, NY
• eBook Packages
• Hardcover ISBN 978-0-387-90340-8
• Softcover ISBN 978-1-4613-9440-2
• eBook ISBN 978-1-4613-9438-9
• Series ISSN 0172-6056
• Edition Number 1
• Number of Pages , 534
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site