Geometry of Convex Sets

333 Pages·2015·1.713 MB·English

by  Leonard, I. Ed., Lewis, James Edward

GEOMETRY OF CONVEX SETS

GEOMETRY OF
CONVEX SETS
I. E. LEONARD
Department of Mathematical and Statistical Sciences
University of Alberta, Edmonton, AB, Canada
J. E. LEWIS
Department of Mathematical and Statistical Sciences
University of Alberta, Edmonton, AB, Canada
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LibraryofCongressCataloging-in-PublicationData:
Leonard,I.Ed.,1938-
Geometryofconvexsets/I.E.Leonard,J.E.Lewis.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-1-119-02266-4(cloth)
1. Convexsets. 2. Geometry. I.Lewis,J.E.(JamesEdward)II.Title.
QA640.L462016
516(cid:2).08–dc23
2015021566
CoverimagecourtesyofI.E.Leonard
Setin10/12ptTeXGyreTermesbySPiGlobal,Chennai,India
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
1 2016
Contents
Preface ix
1 IntroductiontoN-DimensionalGeometry 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
vi Contents
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 Rn 108
2.5.1 Sequences and Compact Sets in Rn 112
2.5.2 Completeness 115
2.6 Applications of Compactness 117
2.6.1 Continuous Functions 117
2.6.2 Equivalent Norms on Rn 119
2.6.3 Distance between Sets in Rn 121
2.6.4 Support Hyperplanes for Compact Sets in Rn 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
Contents vii
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’sTheorem 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