## Join Geometries: A Theory of Convex Sets and Linear GeometryThe 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. |

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### Contents

The Abstract Theory of Join Operations | 46 |

The Generation of Convex SetsConvex Hulls | 124 |

The Operation of Extension | 156 |

Copyright | |

10 other sections not shown

### Other editions - View all

Join Geometries: A Theory of Convex Sets and Linear Geometry W. Prenowitz,J. Jantosciak Limited preview - 2012 |

Join Geometries: A Theory of Convex Sets and Linear Geometry W. Prenowitz,J. Jantosciak No preview available - 2011 |

### Common terms and phrases

3-space analogue answer Apply associative assume basic called Chapter closed closure Compare Exercise component condition Consider contains Conversely convex hull convex set Corollary correspondence covers defined definition denoted determined distinct elements end of Section endpoint equivalent Euclidean geometry example Exercise exists expressed extension extreme point extreme set face familiar Figure Finally finite formal formula geometry let given halfspace Hence holds idea implies indicated interior intersection join geometry join operation L-ray least linear set linear space linearly independent nonempty Note notion o-rays Observe obtain opposite ordered geometry pair plane points polytope postulates principle PROOF proper Prove rank region relation Remark result satisfies segment separates set of points Similarly subset Suppose Theorem theory union unique Verify vertices yields