Foundations of Convex Geometry (Australian Mathematical Society Lecture Series, Series Number 12)

This book on the foundations of Euclidean geometry aims to present the subject from the point of view of present day mathematics, taking advantage of all the developments since the appearance of Hilbert's classic work. Here real affine space is characterised by a small number of axioms involving points and line segments making the treatment self-contained and thorough, many results being established under weaker hypotheses than usual. The treatment should be totally accessible for final year undergraduates and graduate students, and can also serve as an introduction to other areas of mathematics such as matroids and antimatroids, combinatorial convexity, the theory of polytopes, projective geometry and functional analysis. Excellent author Based on graduate courses No competition at this level Coppel, W. A.Foundations of convex geometry (Australian MS lecture series,vol.12)(CUP,1998)(ISBN 0521639700)(600dpi)(236p) ......Page 4 Copyright ......Page 5 Contents ......Page 7 AUSTRALIAN MATHEMATICAL SOCIETY LECTURE SERIES ......Page 3 Preface XI ......Page 10 Introduction 1 ......Page 14 1 Partially ordered sets 3 ......Page 16 2 Aligned spaces 6 ......Page 19 3 Anti-exchange alignments 14 ......Page 27 4 Antimatroids 16 ......Page 29 5 Exchange alignments 20 ......Page 33 6 Notes 24 ......Page 37 1 Examples 27 ......Page 40 2 Convex geometries 31 ......Page 44 3 Additional axioms 36 ......Page 49 4 Examples (continued) 43 ......Page 56 5 Notes 47 ......Page 60 1 Lines 49 ......Page 62 2 Linear geometries 54 ......Page 67 3 Helly's theorem and its relatives 66 ......Page 79 1 Faces 71 ......Page 84 2 Polytopes 74 ......Page 87 3 Factor geometries 81 ......Page 94 4 Notes 84 ......Page 97 1 Dense linear geometries 85 ......Page 98 2 Unending linear geometries 95 ......Page 108 3 Cones 103 ......Page 116 4 Polyhedra 109 ......Page 122 5 Notes 114 ......Page 127 1 Introduction 115 ......Page 128 2 Projective geometry 119 ......Page 132 3 The Desargues property 124 ......Page 137 4 Factor geometries 131 ......Page 144 5 Notes 135 ......Page 148 1 Coordinates 137 ......Page 150 2 Isomorphisms 143 ......Page 156 3 Vector sums 150 ......Page 163 4 Notes 154 ......Page 167 1 Introduction 157 ......Page 170 2 Separation properties 164 ......Page 177 3 Fundamental theorem of ordered geometry 172 ......Page 185 4 Metric and norm 178 ......Page 191 5 Notes 182 ......Page 195 1 The Hausdorff metric 185 ......Page 198 2 The space ^(X) 189 ......Page 202 3 Embeddings of %(X) 192 ......Page 205 4 The Krein-Kakutani theorem 196 ......Page 209 5 Notes 209 ......Page 222 References 211 ......Page 224 Notations 216 ......Page 229 Axioms 217 ......Page 230 Index 218 ......Page 231 cover......Page 1

This book presents the foundations of Euclidean geometry from the point of view of mathematics, taking advantage of all the developments since the appearance of Hilbert's classic work. Here, real affine space is characterized by a small number of axioms involving points and line segments making the treatment self-contained and thorough. This treatment is accessible for final year undergraduates and graduate students, and can also serve as an introduction to other areas of mathematics such as matroids and antimatroids, combinatorial convexity, the theory of polytopes, projective geometry and functional analysis.