Metric Embeddings: Bilipschitz and Coarse Embeddings into Banach Spaces
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
De Gruyter
2013
|
| Schriftenreihe: | De Gruyter Studies in Mathematics
49 |
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. [335] - 360 |
| Beschreibung: | XI, 372 S. 240 mm x 170 mm |
| ISBN: | 3110263408 9783110263404 9783119166225 |
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IMAGE 1
CONTENTS
PREFACE V
1 INTRODUCTION: EXAMPLES OF METRICS, EMBEDDINGS, AND APPLICATIONS 1
1.1 METRIC SPACES: DEFINITIONS AND MAIN EXAMPLES 1
1.2 TYPES OF EMBEDDINGS: ISOMETRIC, BILIPSCHITZ, COARSE, AND UNIFORM . .
. . 6 1.2.1 ISOMETRIC EMBEDDINGS 6
1.2.2 BILIPSCHITZ EMBEDDINGS 9
1.2.3 COARSE AND UNIFORM EMBEDDINGS 17
1.3 PROBABILITY THEORY TERMINOLOGY AND NOTATION 22
1.4 APPLICATIONS TO THE SPARSEST CUT PROBLEM 25
1.5 EXERCISES 26
1.6 NOTES AND REMARKS 27
1.6.1 TO SECTION 1.1 27
1.6.2 TO SECTION 1.2 28
1.6.3 TO SECTION 1.3 29
1.6.4 TO SECTION 1.4 30
1.6.5 TO EXERCISES 30
1.7 ON APPLICATIONS IN TOPOLOGY 30
1.8 HINTS TO EXERCISES 32
2 EMBEDDABILITV OF LOCALLY FINITE METRIC SPACES INTO BANACH SPACES IS
FINITELY DETERMINED. RELATED BANACH SPACE THEORY 34
2.1 INTRODUCTION 34
2.2 BANACH SPACE THEORY: ULTRAFILTERS, ULTRAPRODUCTS, FINITE
REPRESENTABILITY . 35 2.2.1 ULTRAFILTERS 35
2.2.2 ULTRAPRODUCTS 37
2.2.3 FINITE REPRESENTABILITY 40
2.3 PROOFS OF THE MAIN RESULTS ON RELATIONS BETWEEN EMBEDDABILITY OF A
LOCALLY FINITE METRIC SPACE AND ITS FINITE SUBSETS 44
2.3.1 PROOF IN THE BILIPSCHITZ CASE 44
2.3.2 PROOF IN THE COARSE CASE 52
2.3.3 REMARKS ON EXTENSIONS OF FINITE DETERMINATION RESULTS 53
HTTP://D-NB.INFO/1033298514
IMAGE 2
VIII
CONTENTS
2.4 BANACH SPACE THEORY: TYPE AND COTYPE OF BANACH SPACES, KHINCHIN AND
KAHANE INEQUALITIES 53
2.4.1 RADEMACHER TYPE AND COTYPE 53
2.4.2 KAHANE-KHINCHIN INEQUALITY 57
2.4.3 CHARACTERIZATION OF SPACES WITH TRIVIAL TYPE OR COTYPE 66
2.5 SOME COROLLARIES OF THE THEOREMS ON FINITE DETERMINATION OF
EMBEDDABILITY OF LOCALLY FINITE METRIC SPACES 75
2.6 EXERCISES 76
2.7 NOTES AND REMARKS 77
2.8 HINTS TO EXERCISES 79
3 CONSTRUCTIONS OF EMBEDDINGS 80
3.1 PADDED DECOMPOSITIONS AND THEIR APPLICATIONS TO CONSTRUCTIONS OF
EMBEDDINGS 80
3.2 PADDED DECOMPOSITIONS OF MINOR-EXCLUDED GRAPHS 84
3.3 PADDED DECOMPOSITIONS IN TERMS OF BALL GROWTH 90
3.4 GLUING SINGLE-SCALE EMBEDDINGS 93
3.5 EXERCISES 102
3.6 NOTES AND REMARKS 102
3.7 HINTS TO EXERCISES 104
4 OBSTACLES FOR EMBEDDABILITY: POINCARE INEQUALITIES 105
4.1 DEFINITION OF POINCARE INEQUALITIES FOR METRIC SPACES 105
4.2 POINCARE INEQUALITIES FOR EXPANDERS 107
4.3 LP-DISTORTION IN TERMS OF CONSTANTS IN POINCARE INEQUALITIES 112
4.4 EUCLIDEAN DISTORTION AND POSITIVE SEMIDEFINITE MATRICES 114
4.5 FOURIER ANALYTIC METHOD OF GETTING POINCARE INEQUALITIES 116
4.6 EXERCISES 127
4.7 NOTES AND REMARKS 128
4.8 A BIT OF HISTORY OF COARSE EMBEDDABILITY 129
4.9 HINTS TO EXERCISES 130
5 FAMILIES OF EXPANDERS AND OF GRAPHS WITH LARGE GIRTH 131
5.1 INTRODUCTION 131
5.2 SPECTRAL CHARACTERIZATION OF EXPANDERS 132
5.3 KAZHDAN'S PROPERTY (T) AND EXPANDERS 137
IMAGE 3
CONTENTS I X
5.4 GROUPS WITH PROPERTY (T) 142
5.4.1 FINITE GENERATION OF S L N ( L ) 143
5.4.2 FINITE QUOTIENTS OF S L N ( Z ) 144
5.4.3 PROPERTY (T) FOR GROUPS S L N ( Z ) 145
5.4.4 CRITERION FOR PROPERTY (T) 145
5.5 ZIGZAG PRODUCTS 146
5.6 GRAPHS WITH LARGE GIRTH: BASIC DEFINITIONS 155
5.7 GRAPH LIFT CONSTRUCTIONS AND 11-EMBEDDABLE GRAPHS WITH LARGE GIRTH
156
5.8 PROBABILISTIC PROOF OF EXISTENCE OF EXPANDERS 164
5.9 SIZE AND DIAMETER OF GRAPHS WITH LARGE GIRTH: BASIC FACTS 167
5.10 RANDOM CONSTRUCTIONS OF GRAPHS WITH LARGE GIRTH 169
5.11 GRAPHS WITH LARGE GIRTH USING VARIATIONAL TECHNIQUES 170
5.12 INEQUALITIES FOR THE SPECTRAL GAP OF GRAPHS WITH LARGE GIRTH 174
5.13 BIGGS'S CONSTRUCTION OF GRAPHS WITH LARGE GIRTH 175
5.14 MARGULIS'S 1982 CONSTRUCTION OF GRAPHS WITH LARGE GIRTH 177
5.15 FAMILIES OF EXPANDERS WHICH ARE NOT COARSELY EMBEDDABLE ONE INTO
ANOTHER 178
5.16 EXERCISES 181
5.17 NOTES AND REMARKS 183
5.17.1 BOUNDS FOR SPECTRAL GAPS 187
5.17.2 GRAPHS WITH VERY LARGE SPECTRAL GAPS 187
5.17.3 SOME MORE RESULTS AND CONSTRUCTIONS 188
5.18 HINTS TO EXERCISES 189
6 BANACH SPACES WHICH DO NOT ADMIT UNIFORMLY COARSE EMBEDDINGS OF
EXPANDERS 191
6.1 BANACH SPACES WHOSE BALLS ADMIT UNIFORM EMBEDDINGS INTO L I 192
6.2 BANACH SPACES NOT ADMITTING COARSE EMBEDDINGS OF EXPANDER FAMILIES,
USING INTERPOLATION 195
6.3 BANACH SPACE THEORY: A CHARACTERIZATION OF REFLEXIVITY 200
6.4 SOME CLASSES OF SPACES WHOSE BALLS ARE NOT UNIFORMLY EMBEDDABLE INTO
L I 204
6.4.1 STABLE METRIC SPACES AND ITERATED LIMITS 204
6.4.2 NON-EMBEDDABILITY RESULT 206
6.5 EXAMPLES OF NON-REFLEXIVE SPACES WITH NONTRIVIAL TYPE 208
6.6 EXERCISES 215
IMAGE 4
X
CONTENTS
6.7 NOTES AND REMARKS 215
6.8 HINTS TO EXERCISES 217
7 STRUCTURE PROPERTIES OF SPACES WHICH ARE NOT COARSELY EMBEDDABLE INTO
A HILBERT SPACE 218
7.1 EXPANDER-LIKE STRUCTURES IMPLYING COARSE NON-EMBEDDABILITY INTO L \
218
7.2 ON THE STRUCTURE OF LOCALLY FINITE SPACES WHICH DO NOT ADMIT COARSE
EMBEDDINGS INTO A HILBERT SPACE 220
7.3 EXPANSION PROPERTIES OF METRIC SPACES NOT ADMITTING A COARSE
EMBEDDING INTO A HILBERT SPACE 223
7.4 EXERCISES 226
7.5 NOTES AND REMARKS 226
7.6 HINTS TO EXERCISES 227
8 APPLICATIONS OF MARKOV CHAINS TO EMBEDDABILITY PROBLEMS 228
8.1 BASIC DEFINITIONS AND RESULTS ON FINITE MARKOV CHAINS 228
8.2 MARKOV TYPE 230
8.3 FIRST APPLICATION OF MARKOV TYPE TO EMBEDDABILITY PROBLEMS:
EUCLIDEAN DISTORTION OF GRAPHS WITH LARGE GIRTH 232
8.4 BANACH SPACE THEORY: RENORMINGS OF SUPERREFLEXIVE SPACES,
^-CONVEXITY AND /7-SMOOTHNESS 233
8.4.1 DEFINITIONS AND DUALITY 233
8.4.2 PISIER THEOREM ON RENORMINGS OF UNIFORMLY CONVEX SPACES . . . 239
8.5 MARKOV TYPE OF UNIFORMLY SMOOTH BANACH SPACES 253
8.6 APPLICATIONS OF MARKOV TYPE TO LOWER ESTIMATES OF DISTORTIONS OF
EMBEDDINGS INTO UNIFORMLY SMOOTH BANACH SPACES 259
8.7 EXERCISES 261
8.8 NOTES AND REMARKS 262
8.9 HINTS TO EXERCISES 264
9 METRIC CHARACTERIZATIONS OF CLASSES OF BANACH SPACES 265
9.1 INTRODUCTION 265
9.2 PROOF OF THE RIBE THEOREM THROUGH BOURGAIN'S DISCRETIZATION THEOREM
266 9.2.1 PROVING BOURGAIN'S DISCRETIZATION THEOREM. PRELIMINARY STEP:
IT SUFFICES TO CONSIDER SPACES WITH DIFFERENTIABLE NORM 268 9.2.2 FIRST
STEP: PICKING THE SYSTEM OF COORDINATES 269
9.2.3 SECOND STEP: CONSTRUCTION OF A LIPSCHITZ ALMOST-EXTENSION . . . .
271
IMAGE 5
CONTENTS
X I
9.2.4 THIRD STEP: FURTHER SMOOTHING OF THE MAP USING POISSON KERNELS 276
9.2.5 POISSON KERNEL ESTIMATES AND PROOFS OF LEMMAS 9.14 AND 9.15 283
9.3 TEST-SPACE CHARACTERIZATIONS 287
9.3.1 MORE BANACH SPACE THEORY: SUPERREFLEXIVITY 289
9.3.2 CHARACTERIZATION OF SUPERREFLEXIVITY IN TERMS OF DIAMOND GRAPHS
297
9.4 EXERCISES 302
9.5 NOTES AND REMARKS 303
9.5.1 ANOTHER TEST-SPACE CHARACTERIZATION OF SUPERREFLEXIVITY: BINARY
TREES 305
9.5.2 FURTHER RESULTS ON TEST-SPACES 306
9.5.3 FURTHER RESULTS ON THE RIBE PROGRAM 307
9.5.4 NON-LOCAL PROPERTIES 307
9.6 HINTS TO EXERCISES 307
10 LIPSCHITZ FREE SPACES 308
10.1 INTRODUCTORY REMARKS 308
10.2 LIPSCHITZ FREE SPACES: DEFINITION AND PROPERTIES 308
10.3 THE CASE WHERE DX IS A GRAPH DISTANCE 312
10.4 LIPSCHITZ FREE SPACES OF SOME FINITE METRIC SPACES 317
10.5 EXERCISES 326
10.6 NOTES AND REMARKS 326
10.7 HINTS TO EXERCISES 327
11 OPEN PROBLEMS 328
11.1 EMBEDDABILITY OF EXPANDERS INTO BANACH SPACES 328
11.2 OBSTACLES FOR COARSE EMBEDDABILITY OF SPACES WITH BOUNDED GEOMETRY
INTO A HILBERT SPACE 330
11.2.1 THE MAIN PROBLEM 330
11.2.2 COMMENTS 331
11.3 EMBEDDABILITY OF GRAPHS WITH LARGE GIRTH 332
11.4 COARSE EMBEDDABILITY OF A HILBERT SPACE INTO BANACH SPACES 333
BIBLIOGRAPHY 335
AUTHOR INDEX 361
SUBJECT INDEX 367 |
| any_adam_object | 1 |
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| physical | XI, 372 S. 240 mm x 170 mm |
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| spelling | Ostrovskii, Mikhail I. Verfasser (DE-588)1036534936 aut Metric Embeddings Bilipschitz and Coarse Embeddings into Banach Spaces Mikhail I. Ostrovskii Berlin [u.a.] De Gruyter 2013 XI, 372 S. 240 mm x 170 mm txt rdacontent n rdamedia nc rdacarrier De Gruyter Studies in Mathematics 49 Literaturverz. S. [335] - 360 Banach-Raum (DE-588)4004402-6 gnd rswk-swf Diskreter metrischer Raum (DE-588)4587633-2 gnd rswk-swf Einbettung Mathematik (DE-588)4151233-9 gnd rswk-swf Einbettung Mathematik (DE-588)4151233-9 s Diskreter metrischer Raum (DE-588)4587633-2 s Banach-Raum (DE-588)4004402-6 s DE-604 Erscheint auch als Online-Ausgabe 978-3-11-026401-2 De Gruyter Studies in Mathematics 49 (DE-604)BV000005407 49 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4294310&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026081863&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ostrovskii, Mikhail I. Metric Embeddings Bilipschitz and Coarse Embeddings into Banach Spaces De Gruyter Studies in Mathematics Banach-Raum (DE-588)4004402-6 gnd Diskreter metrischer Raum (DE-588)4587633-2 gnd Einbettung Mathematik (DE-588)4151233-9 gnd |
| subject_GND | (DE-588)4004402-6 (DE-588)4587633-2 (DE-588)4151233-9 |
| title | Metric Embeddings Bilipschitz and Coarse Embeddings into Banach Spaces |
| title_auth | Metric Embeddings Bilipschitz and Coarse Embeddings into Banach Spaces |
| title_exact_search | Metric Embeddings Bilipschitz and Coarse Embeddings into Banach Spaces |
| title_full | Metric Embeddings Bilipschitz and Coarse Embeddings into Banach Spaces Mikhail I. Ostrovskii |
| title_fullStr | Metric Embeddings Bilipschitz and Coarse Embeddings into Banach Spaces Mikhail I. Ostrovskii |
| title_full_unstemmed | Metric Embeddings Bilipschitz and Coarse Embeddings into Banach Spaces Mikhail I. Ostrovskii |
| title_short | Metric Embeddings |
| title_sort | metric embeddings bilipschitz and coarse embeddings into banach spaces |
| title_sub | Bilipschitz and Coarse Embeddings into Banach Spaces |
| topic | Banach-Raum (DE-588)4004402-6 gnd Diskreter metrischer Raum (DE-588)4587633-2 gnd Einbettung Mathematik (DE-588)4151233-9 gnd |
| topic_facet | Banach-Raum Diskreter metrischer Raum Einbettung Mathematik |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4294310&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026081863&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005407 |
| work_keys_str_mv | AT ostrovskiimikhaili metricembeddingsbilipschitzandcoarseembeddingsintobanachspaces |