Geometry of Convex Sets (eBook)

I. E. Leonard, PhD, was a contract lecturer in the Department of Mathematical and Statistical Sciences at the University of Alberta. The author of over 15 peer-reviewed journal articles, he is a technical editor for theCanadian Applied Mathematical Quarterly journal. J. E. Lewis, PhD, is Professor Emeritus in the Department of Mathematical Sciences at the University of Alberta. He was the recipient of the Faculty of Science Award for Excellence in Teaching in 2004 as well as the PIMS Education Prize in 2002.

Preface ix

1 Introduction to N-Dimensional Geometry 1

1.1 Figures in N-Dimensions 1

1.2 Points, Vectors, and Parallel Lines 2

1.2.1 Points and Vectors 2

1.2.2 Lines 4

1.2.3 Segments 11

1.2.4 Examples 12

1.2.5 Problems 18

1.3 Distance in N-Space 19

1.3.1 Metrics 19

1.3.2 Norms 20

1.3.3 Balls and Spheres 23

1.4 Inner Product and Orthogonality 29

1.4.1 Nearest Points 32

1.4.2 Cauchy-Schwarz Inequality 36

1.4.3 Problems 41

1.5 Convex Sets 41

1.6 Hyperplanes and Linear Functionals 45

1.6.1 Linear Functionals 45

1.6.2 Hyperplanes 52

1.6.3 Problems 66

2 Topology 69

2.1 Introduction 69

2.2 Interior Points and Open Sets 72

2.2.1 Properties of Open Sets 80

2.3 Accumulation Points and Closed Sets 83

2.3.1 Properties of Closed Sets 88

2.3.2 Boundary Points and Closed Sets 88

2.3.3 Closure of a Set 90

2.3.4 Problems 92

2.4 Compact Sets in R 94

2.4.1 Basic Properties of Compact Sets 99

2.4.2 Sequences and Compact Sets in R 104

2.4.3 Completeness 106

2.5 Compact Sets in R n 108

2.5.1 Sequences and Compact Sets in R n 112

2.5.2 Completeness 115

2.6 Applications of Compactness 117

2.6.1 Continuous Functions 117

2.6.2 Equivalent Norms on R n 119

2.6.3 Distance between Sets in R n 121

2.6.4 Support Hyperplanes for Compact Sets in R n 127

2.6.5 Problems 130

3 Convexity 135

3.1 Introduction 135

3.2 Basic Properties of Convex Sets 137

3.2.1 Problems 144

3.3 Convex Hulls 146

3.3.1 Problems 155

3.4 Interior and Closure of Convex Sets 157

3.4.1 The Closed Convex Hull 161

3.4.2 Accessibility Lemma 162

3.4.3 Regularity of Convex Sets 164

3.4.4 Problems 169

3.5 Affine Hulls 170

3.5.1 Flats or Affine Subspaces 170

3.5.2 Properties of Flats 172

3.5.3 Affine Basis 173

3.5.4 Problems 178

3.6 Separation Theorems 180

3.6.1 Applications of the Separation Theorem 192

3.6.2 Problems 196

3.7 Extreme Points of Convex Sets 199

3.7.1 Supporting Hyperplanes and Extreme Points 199

3.7.2 Existence of Extreme Points 203

3.7.3 The Krein-Milman Theorem 205

3.7.4 Examples 207

3.7.5 Polyhedral Sets and Polytopes 210

3.7.6 Birkhoff's Theorem 220

3.7.7 Problems 224

4 Helly's Theorem 227

4.1 Finite Intersection Property 227

4.1.1 The Finite Intersection Property 227

4.1.2 Problems 229

4.2 Helly's Theorem 230

4.3 Applications of Helly's Theorem 235

4.3.1 The Art Gallery Theorem 235

4.3.2 Vincensini's Problem 242

4.3.3 Hadwiger's Theorem 249

4.3.4 Theorems of Radon and Carathéodory 257

4.3.5 Kirchberger's Theorem 260

4.3.6 Helly-type Theorems for Circles 262

4.3.7 Covering Problems 266

4.3.8 Piercing Problems 274

4.3.9 Problems 276

4.4 Sets of Constant Width 277

4.4.1 Reuleaux Triangles 277

4.4.2 Properties of Sets of Constant Width 279

4.4.3 Adjunction Complete Convex Sets 285

4.4.4 Sets of Constant Width in the Plane 293

4.4.5 Barbier's Theorem 294

4.4.6 Constructing Sets of Constant Width 297

4.4.7 Borsuk's Problem 304

4.4.8 Problems 309

Bibliography 311

Index 317

"This book presents a very friendly introduction to the basic concepts of classical convex geometry. The book is designed for a one-semester upper undergraduate course and is oriented towards students studying education, engineering and arts. [It] is very well thought out and planned, and contains a lot of interesting and helpful exercises." (Mathematical Reviews/MathSciNet June 2017)

"A good book on the geometry of convex sets in n-dimensional space is elaborated to be used for students in fields touching pure and applied mathematics: education, arts, engineering." (Zentralblatt Math, 2016)