Metric Spaces, Convexity and Nonpositive Curvature (Irma Lectures in Mathematics and Theoretical Physics, Vol. 6)

Preface......Page 7
Contents......Page 9
The work of Hadamard......Page 13
The works of Menger and Wald......Page 16
The works of Busemann and Alexandrov......Page 18
Convexity......Page 20
Lengths of paths in metric spaces......Page 22
The length of a path......Page 23
Arclength as parameter......Page 31
Differentiable paths in Euclidean space......Page 34
The space of paths......Page 36
Notes on Chapter 1......Page 42
Length spaces and geodesic spaces......Page 46
Length spaces......Page 47
Geodesics......Page 62
Limits of geodesics......Page 68
Geodesic spaces......Page 70
Geodesic convexity......Page 79
Menger convexity......Page 81
Notes on Chapter 2......Page 88
Maps between metric spaces......Page 91
K-Lipschitz maps and K-length-non-increasing maps......Page 92
Non-expanding maps......Page 94
Distance non-decreasing maps......Page 99
Local isometries......Page 101
Covering spaces......Page 108
Notes on Chapter 3......Page 112
Distances......Page 115
The Hausdorff distance......Page 117
The Busemann–Hausdorff distance......Page 122
Closed limits of subsets......Page 124
Metrics on the isometry group......Page 131
Notes on Chapter 4......Page 135
Affinely convex subsets......Page 139
Convex hull......Page 145
Convexity in normed vector spaces......Page 149
Limits of convex sets......Page 154
Minkowski's construction......Page 156
The Hilbert geometry......Page 160
Notes on Chapter 5......Page 166
Convex functions......Page 171
Convex functions......Page 172
Convex functions of one variable......Page 182
Notes on Chapter 6......Page 187
Strictly convex normed vector spaces......Page 190
Uniquely geodesic spaces......Page 192
Inner products and p norms......Page 197
Notes on Chapter 7......Page 198
Busemann spaces......Page 199
Local geodesics in Busemann spaces......Page 208
Geodesic convexity in Busemann spaces......Page 210
Convex functions on Busemann spaces......Page 211
Notes on Chapter 8......Page 215
Locally convex spaces......Page 222
Locally convex spaces......Page 223
Variation of local geodesics......Page 230
The universal covering of a locally convex metric space......Page 236
Notes on Chapter 9......Page 238
Asymptotic rays and the visual boundary......Page 241
Asymptotic rays......Page 242
The visual boundary......Page 247
Notes on Chapter 10......Page 251
Isometries......Page 253
Minimal displacement, minimal sets and the isometry types......Page 254
Axial isometries......Page 261
Periodic geodesics......Page 265
Axial isometries of Busemann spaces......Page 268
Parallel lines......Page 270
Notes on Chapter 11......Page 272
Busemann functions......Page 273
Co-rays......Page 277
Horospheres......Page 281
Notes on Chapter 12......Page 284
References......Page 287
Index......Page 295