Minkowski geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2004
|
| Edition: | transferred to digital print. |
| Series: | Encyclopedia of mathematics and its applications
63 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 346 S. Ill., graph. Darst. |
| ISBN: | 052140472X 9780521404723 |
Staff View
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| 245 | 1 | 0 | |a Minkowski geometry |c A. C. Thompson |
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| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2004 | |
| 300 | |a XVI, 346 S. |b Ill., graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804138033304305664 |
|---|---|
| adam_text | CONTENTS
Preface
page
ix
Acknowledgements
XV
0
The algebraic properties of linear spaces and convex sets
1
0.1
Linear spaces
1
0.2
Convex sets
6
0.3
Notes
9
1
Norms and norm topologies
13
1.1
Norm topologies
14
1.2
The unique linear topology on Rrf
27
1.3
The Hahn-Banach theorem
32
1.4
The existence and uniqueness of
Haar
measure
36
1.5
Notes
42
2
Convex bodies
45
2.1
Separation and support theorems
46
2.2
Support functions and polar reciprocals
48
2.3
Volumes and mixed volumes
53
2.4
Various derived metrics
60
2.5
Approximation of convex sets and the Blaschke selection
theorem
64
2.6
Notes
71
3
Comparisons and contrasts with Euclidean space
75
3.1
The
Mazur—
Ulam
theorem
76
3.2
Normality in Minkowski space
77
3.3
The
Löwner
ellipsoid
80
3.4
Characterizations of Euclidean space
85
3.5
Notes
94
Contents
Two-dimensional
Minkowski
spaces
99
4.1
Inscribed regular hexagons and other constructions
100
4.2
Sets of constant width and equichordal sets
106
4.3
Lengths of curves, perimeter of the unit ball
111
4.4
The isoperimetric problem in a Minkowski plane
118
4.5
Isoperimetric inequalities
123
4.6
Transversality
125
4.7
Radon curves
127
4.8
Notes
129
The concept of area and content
135
5.1
Requirements and examples
137
5.2
The role of the function
σΒ
141
5.3
The properties and the normalization of I
145
5.4
The isoperimetrices that arise from Examples
5.1.4 150
5.5
Further properties of I
171
5.6
Notes
182
Special properties of the Holmes-Thompson definition
187
6.1
The convexity of the area function
σ
187
6.2
Properties of the mapping I
195
6.3
Cauchy s formula for surface areas
201
6.4
Integral geometry in Minkowski spaces
205
6.5
Bounds for the surface area of
В
212
6.6
Miscellaneous properties
215
6.7
Notes
222
Special properties of the Busemann definition
229
7.1
The convexity of the area function
σ
229
7.2
Properties of the mapping I
233
7.3
Area and Hausdorff measures
237
7.4
Bounds for the surface area of
В
242
7.5
Notes
245
Trigonometry
251
8.1
The functions cm and sm
251
8.2
The function a
258
8.3
Trigonometric formulas
260
8.4
Differentiation of the trigonometric functions
264
8.5
Notes
271
Various numerical parameters
275
9.1
Projection constants
276
9.2
Macphail s constant
283
Contents
9.3 The inner
metric
286
9.4
The girth, perimeter, inner radius and inner diameter of X
288
9.5
Five examples in Rd
293
9.6
Relationships with the Banach-Mazur distance and extreme
values
300
9.7
Notes
304
10
Fifty problems
307
References
313
Notation index
331
Author index
335
Subject index
339
|
| adam_txt |
CONTENTS
Preface
page
ix
Acknowledgements
XV
0
The algebraic properties of linear spaces and convex sets
1
0.1
Linear spaces
1
0.2
Convex sets
6
0.3
Notes
9
1
Norms and norm topologies
13
1.1
Norm topologies
14
1.2
The unique linear topology on Rrf
27
1.3
The Hahn-Banach theorem
32
1.4
The existence and uniqueness of
Haar
measure
36
1.5
Notes
42
2
Convex bodies
45
2.1
Separation and support theorems
46
2.2
Support functions and polar reciprocals
48
2.3
Volumes and mixed volumes
53
2.4
Various derived metrics
60
2.5
Approximation of convex sets and the Blaschke selection
theorem
64
2.6
Notes
71
3
Comparisons and contrasts with Euclidean space
75
3.1
The
Mazur—
Ulam
theorem
76
3.2
Normality in Minkowski space
77
3.3
The
Löwner
ellipsoid
80
3.4
Characterizations of Euclidean space
85
3.5
Notes
94
Contents
Two-dimensional
Minkowski
spaces
99
4.1
Inscribed regular hexagons and other constructions
100
4.2
Sets of constant width and equichordal sets
106
4.3
Lengths of curves, perimeter of the unit ball
111
4.4
The isoperimetric problem in a Minkowski plane
118
4.5
Isoperimetric inequalities
123
4.6
Transversality
125
4.7
Radon curves
127
4.8
Notes
129
The concept of area and content
135
5.1
Requirements and examples
137
5.2
The role of the function
σΒ
141
5.3
The properties and the normalization of I
145
5.4
The isoperimetrices that arise from Examples
5.1.4 150
5.5
Further properties of I
171
5.6
Notes
182
Special properties of the Holmes-Thompson definition
187
6.1
The convexity of the area function
σ
187
6.2
Properties of the mapping I
195
6.3
Cauchy's formula for surface areas
201
6.4
Integral geometry in Minkowski spaces
205
6.5
Bounds for the surface area of
В
212
6.6
Miscellaneous properties
215
6.7
Notes
222
Special properties of the Busemann definition
229
7.1
The convexity of the area function
σ
229
7.2
Properties of the mapping I
233
7.3
Area and Hausdorff measures
237
7.4
Bounds for the surface area of
В
242
7.5
Notes
245
Trigonometry
251
8.1
The functions cm and sm
251
8.2
The function a
258
8.3
Trigonometric formulas
260
8.4
Differentiation of the trigonometric functions
264
8.5
Notes
271
Various numerical parameters
275
9.1
Projection constants
276
9.2
Macphail's constant
283
Contents
9.3 The inner
metric
286
9.4
The girth, perimeter, inner radius and inner diameter of X
288
9.5
Five examples in Rd
293
9.6
Relationships with the Banach-Mazur distance and extreme
values
300
9.7
Notes
304
10
Fifty problems
307
References
313
Notation index
331
Author index
335
Subject index
339 |
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| illustrated | Illustrated |
| index_date | 2024-07-02T22:07:16Z |
| indexdate | 2024-07-09T21:21:45Z |
| institution | BVB |
| isbn | 052140472X 9780521404723 |
| language | English |
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| physical | XVI, 346 S. Ill., graph. Darst. |
| publishDate | 2004 |
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| publishDateSort | 2004 |
| publisher | Cambridge Univ. Press |
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| series | Encyclopedia of mathematics and its applications |
| series2 | Encyclopedia of mathematics and its applications |
| spelling | Thompson, Anthony C. Verfasser aut Minkowski geometry A. C. Thompson transferred to digital print. Cambridge [u.a.] Cambridge Univ. Press 2004 XVI, 346 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 63 Geometrie der Zahlen (DE-588)4227477-1 gnd rswk-swf Geometrie der Zahlen (DE-588)4227477-1 s DE-604 Encyclopedia of mathematics and its applications 63 (DE-604)BV000903719 63 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016749248&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Thompson, Anthony C. Minkowski geometry Encyclopedia of mathematics and its applications Geometrie der Zahlen (DE-588)4227477-1 gnd |
| subject_GND | (DE-588)4227477-1 |
| title | Minkowski geometry |
| title_auth | Minkowski geometry |
| title_exact_search | Minkowski geometry |
| title_exact_search_txtP | Minkowski geometry |
| title_full | Minkowski geometry A. C. Thompson |
| title_fullStr | Minkowski geometry A. C. Thompson |
| title_full_unstemmed | Minkowski geometry A. C. Thompson |
| title_short | Minkowski geometry |
| title_sort | minkowski geometry |
| topic | Geometrie der Zahlen (DE-588)4227477-1 gnd |
| topic_facet | Geometrie der Zahlen |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016749248&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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