Table of Contents

Preface vii

1 Introduction to N-Dimensional Geometry 1

1.2 Points Vectors and Parallel Lines 1

1.2.5 Problems 1

1.4 Inner Product and Orthogonality 3

1.4.3 Problems 3

1.6 Hyperplanes and Linear Functionals 5

1.6.3 Problems 5

2 Topology 13

2.3 Accumulation Points and Closed Sets 13

2.3.4 Problems 13

2.6 Applications of Compactness 20

2.6.5 Problems 20

3 Convexity 35

3.2 Basic Properties of Convex Sets 35

3.2.1 Problems 35

3.3 Convex Hulls 43

3.3.1 Problems 43

3.4 Interior and Closure of Convex Sets 52

3.4.4 Problems 52

3.5 Affine Hulls 55

3.5.4 Problems 55

3.6 Separation Theorems 66

3.6.2 Problems 66

3.7 Extreme Points of Convex Sets 78

3.7.7 Problems 78

4 Helly's Theorem 89

4.1 Finite Intersection Property 89

4.1.2 Problems 89

4.3 Applications of Helly's Theorem 92

4.3.9 Problems 92

4.4 Sets of Constant Width 99

4.4.8 Problems 99

Bibliography 109

Index 113

About the Author

I. E. Leonard, PhD, is 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 the Canadian Applied Mathematical Quarterly.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.