Join Geometries: A Theory of Convex Sets and Linear Geometry

553 Pages·1979·42.353 MB·English

by  Walter Prenowitz, James Jantosciak (auth.)

Undergraduate Texts in Mathematics
Editors
F. W. Gehring
P. R. Halmos
Advisory Board
C. DePrima
I. Herstein
J. Kiefer
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From THE LABYRINTH (Harper & Brothers). © 1960 Saul Steinberg. Originally in The New
Yorker.
Walter Prenowitz
James Jantosciak
Join Geometries
A Theory of Convex Sets
and Linear Geometry
Springer-Verlag
New York Heidelberg Berlin
Walter Prenowitz (ret.) James Jantosciak
Department of Mathematics Department of Mathematics
Brooklyn College Brooklyn College
City University of New York City University of New York
New York, NY 11210 New York, NY 11210
USA USA
Editorial Board
F. W. Gehring P. R. Halmos
University of Michigan University of Indiana
Department of Mathematics Department of Mathematics
Ann Arbor, Michigan 48104 Bloomington, Indiana 47401
USA USA
AMS Subject Classifications: 50-01, 52-01
With 404 Figures
Library of Congress Cataloging in Publication Data
Prenowitz, Walter, 1906-
Join geometries.
(Undergraduate texts in mathematics)
Includes index.
1. Geometry. I. Jantosciak, James, joint
author. II. Title.
QA447.P73 516 79-1457
All rights reserved.
No part of this book may be translated or reproduced in any form
without written permission from Springer-Verlag.
© 1979 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1st edition 1979
9 8 7 6 543 2 1
ISBN-13:978-1-4613-9440-2 e-ISBN-13:978-1-4613-9438-9
DOl: 10.1007/978-1-4613-9438-9
To Sophie
Contents
Introduction xv
Acknowledgments XXl
1 The Join and Extension Operations in Euclidean Geometry
l.l The Notion of Segment: Closed and Open 2
1.2 The Join of Two Distinct Points 3
1.3 Two Basic Properties of the Join Operation 3
1.4 A Crucial Question 4
1.5 The Join of Two Geometric Figures 5
1.6 Joins of Several Points: Does the Associative Law Hold? 9
1.7 The Join of Two Intersecting Geometric Figures 13
1.8 A Decision Must Be Made 14
1.9 The Join of a Point and Itself 14
1.l0 The Umestricted Applicability of the Join Operation 16
1.11 The Umestricted Validity of the Associative Law for Join 17
1.12 The Universality of the Associative Law 20
1.13 Alternatives to the Definition aa = a 21
1.14 Convex Sets 25
1.15 A Geometric Proof in Join Terminology 28
1.16 A New Operation: The Idea of Extension 33
1.17 The Notion of Halfline or Ray 33
1.18 Formal Definition of the Extension Operation 34
1.19 Identification of Extension as a Geometrical Figure 34
1.20 Properties of the Extension Operation 35
1.21 The Extension of Two Geometric Figures 37
1.22 The Generation of Unbounded or Endless Figures 38
1.23 Is There an Alternative to Join as Open Segment? 42
2 The Abstract Theory of Join Operations 46
2.1 The Join Operation in Euclidean Geometry 46
2.2 Join Operations in a Set-Join Systems 51
2.3 The Postulates for the Operation Join 53
vii
viii Contents
2.4 Application of the Theory to Euclidean Geometry 56
2.5 Elementary Properties of Join 58
2.6 Generalizations of J1-J4 to Sets 63
2.7 Extension of the Join Operation to n Terms 66
2.8 Comparison with the Conventional View of Geometry 68
2.9 Convex Sets 70
2.10 Joins, Intersections and Unions of Convex Sets 74
2.11 Interiors and Frontiers of Convex Sets 78
2.12 Euclidean Interiors and Frontiers 80
2.13 Interiority Properties of Convex Sets &6
2.14 Absorption by Joining 91
2.15 Closures of Euclidean Convex Sets, Intuitively Treated 92
2.16 The Oosure of a Convex Set 95
2.17 Oosure Properties of Convex Sets 97
2.18 Composition of the Interior and Oosure Functions 100
2.19 The Boundary of a Convex Set 104
2.20 Project: Another Formulation of the Theory of Join 106
2.21 What Does the Theory Apply To? 108
2.22 The Triode Model 108
2.23 A Peculiarity of the Triode Model 110
2.24 The Cartesian Join Model 112
2.25 The Metric Interior of a Set in Euclidean Geometry 114
2.26 The Metric Closure of a Set in Euclidean Geometry 120
3 The Generation of Convex Sets-Convex Hulls 124
3.1 Introduction to Convexification: Two Euclidean Examples 124
3.2 Convexification of an Arbitrary Set 125
3.3 Finitely Generated Convex Sets-the Concept of Polytope 132
3.4 A Formula for a Polytope 133
3.5 A Distributive Law 138
3.6 An Absorption Property of Polytopes 139
3.7 Interiors and Oo~ures of Polytopes 140
3.8 Powers of a Set and Polytopes 143
3.9 The Representation of Convex Hulls 145
3.10 Convex Hulls and Powers of a Set 146
3.11 Bounded Sets 147
3.12 Project-the Closed Join Operation 148
3.13 Convex Hulls of Finite Families of Sets-Generalized
Polytopes 149
3.14 Properties of Generalized Polytopes 151
3.15 Examples of Generalized Polytopes 153
4 The Operation of Extension 156
4.1 Definition of the Extension Operation 156
4.2 The Extension Operation for Sets 160
4.3 The Monotonic Law for Extension 163
4.4 Distributive Laws for Extension 164
4.5 The Relation "Intersects" or "Meets" 164
Contents IX
4.6 The Three Term Transposition Law 165
4.7 The Mixed Associative Law 167
4.8 Three New Postulates 168
4.9 Discussion of the Postulates 170
4.10 Formal Consequences of the Postulate Set 11-17 172
4.11 Formal Consequences Continued 174
4.12 Joins and Extensions of Rays 177
4.13 Solving Problems 180
4.14 Extreme Points of Convex Sets 183
4.15 Open Convex Sets 189
4.16 The Intersection of Open Sets 192
4.17 The Join of Open Sets 193
4.18 Segments Are Infinite 194
4.19 The Polytope Interior Theorem 195
4.20 The Interior of a Join of Convex Sets 196
4.21 The Generalized Polytope Interior Theorem 197
4.22 Closed Convex Sets 198
4.23 The Theory of Order 200
4.24 Ordered Sets of Points 202
4.25 Separation of Points by a Point 203
4.26 Perspectivity and Precedence of Points 204
5 Join Geometries 208
5.1 The Concept of a Join Geometry 208
5.2 A List of Join Geometries 211
5.3 Deducibility and Counterexamples 216
5.4 The Existence of Points 218
5.5 Isomorphism of Join Systems 219
5.6 A Gass of Join Geometries of Arbitrary Dimension 226
5.7 IRn Is Converted into a Vector Space 230
5.8 Restatement of the Definition of Join in IRn 232
5.9 Proof That (IRn, .) Is a Join Geometry 233
5.10 Linear Inequalities and Halfspaces 236
5.11 Pathological Convex Sets 240
5.12 An Infinite Dimensional Join Geometry 240
5.13 Three Pathological Convex Sets 241
5.14 Is There a Simple Way to Construct Pathological Convex
Sets? 243
6 Linear Sets 245
6.1 The Notion of Linear Set 245
6.2 The Definition of Linear Set 247
6.3 Conditions for Linearity 248
6.4 Constructing Linear Sets from Linear Sets 249
6.5 The Construction of a Linear Set from a Convex Set 253
6.6 Linear Sets Give Rise to Join Geometries 254
6.7 The Generation of Linear Sets: Two Euclidean Examples 255
6.8 The Generation of Linear Sets: General Case 256
X Contents
6.9 The Linear Hull of a Finite Set: Finitely Generated
Linear Sets 257
6.10 The Definition of Line 258
6.11 The Linear Hull of an Arbitrary Set 260
6.12 The Linear Hull of a Finite Family of Sets 262
6.13 Linear Hulls of Interiors and Closures 264
6.14 Geometric Relations of Points-Linear Dependence and
Independence 265
6.15 Properties of Linearly Independent Points 267
6.16 Simplexes 268
6.17 Linear Dependence and Intersection of Joins 269
6.18 Covering in the Family of Linear Sets-Hyperplanes 272
6.l9 The Height of a Linear Set-Approach to a Theory of
Dimension 274
6.20 Linear Sets and the Interior Operation 276
6.21 Applications: Interiority Properties 278
6.22 The Prevalence of Nonpathological Convex Sets 282
6.23 The Linear Hull of a Pair of Convex Sets 283
6.24 The Interior of the Extension of Two Convex Sets 284
7 Extremal Structure of Convex Sets: Components and Faces 287
7.1 The Notion of an Extreme Set of a Convex Set 287
7.2 The Definition of Extreme Set of a Convex Set 289
7.3 Remarks on the Definition 289
7.4 Elementary Properties of Extreme Sets 291
7.5 The Interior Operation Is Applied to Extreme Sets 293
7.6 The Closure Operation Is Applied to Extreme Sets 294
7.7 Other Characterizations of Extremeness 295
7.8 Is Extremeness Preserved under Join and Extension? 298
7.9 Extreme Sets with a Preassigned Interior Point 300
7.10 Classifying Extreme Sets 302
7.11 Open Extreme S~ts: Components 304
7.12 The Partition Theorem for Convex Sets 305
7.l3 The Component Structure of a Convex Set 306
7.l4 Components as Maximal Open Subsets 307
7.l5 Intersection Properties of Components 308
7.16 Components and Perspectivity of Points 310
7.l7 Components of Polytopes 312
7.18 The Concept of a Face of a Convex Set 315
7.19 The Nonseparation Property of a Face 316
7.20 Elementary Properties of Faces of Convex Sets 318
7.21 Additional Properties of Faces 320
7.22 Facial Structure of Convex Sets 321
7.23 Facial Structure Continued 324
7.24 A Correspondence Between Components and Faces 325
7.25 Faces of Polytopes 326
7.26 Covering in the Family of Faces of a Convex Set 328
7.27 Extreme Sets and Extremal Linear Spaces 332
Contents xi
7.28 Associated Extreme Sets 335
7.29 Covering of Extremals Arising from Faces 336
7.30 Extremal Hyperplanes and Exposed Faces 337
7.31 Supporting and Tangent Hyperplanes 339
8 Rays and Halfspaces 342
8.1 Elementary Properties of Rays 342
8.2 Elementary Operations on Rays 345
8.3 Opposite Rays 348
8.4 Separation of Two Rays by a Common Endpoint 350
8.5 The Partition of Space into Rays 351
8.6 Closed Rays 352
8.7 The Linear Hull of a Ray 353
8.8 The Halfspaces of a Linear Set 355
8.9 A Point of Terminology 357
8.10 Elementary Properties of L-Rays 358
8.11 Elementary Operations on L-Rays 360
8.12 Opposite Halfspaces 361
8.13 Separation of Two Halfspaces by a Common Edge 364
8.14 The Partition of Space into Halfspaces 365
8.15 Closed Halfspaces 366
8.16 The Linear Hull of a Halfspace 366
9 Cones and Hypercones 368
9.1 Cones 368
9.2 Convex Cones 371
9.3 Pointed Convex Cones 373
9.4 Generation of Convex Cones 375
9.5 How Shall Convex o-Cones Be Generated? 377
9.6 Polyhedral Cones 378
9.7 Finiteness of Generation of Convex Cones 380
9.8 Extreme Rays 382
9.9 Extreme Rays of Convex Cones 385
9.10 The Analogous Development Breaks Down 386
9.11 Regularly Imbedded Rays 389
9.12 The Generation of Convex Cones by Extreme Rays 391
9.13 Hypercones 397
9.14 Elementary Properties of Hypercones 398
9.15 Convex Hypercones 398
9.16 Tapered Convex Hypercones 399
9.17 Generation of Convex Hypercones 400
9.18 Polyhedral Hypercones 400
9.19 Finiteness of Generation of Convex Hypercones 401
9.20 Extreme Halfspaces of Convex Hypercones 401
9.21 Regularly Imbedded Halfspaces 403
9.22 The Generation of Convex Hypercones by Extreme
Halfspaces 403