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Thompson A.C. — Minkowski Geometry
Thompson A.C. — Minkowski Geometry

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Название: Minkowski Geometry

Автор: Thompson A.C.

Аннотация:

Minkowski geometry is a non-Euclidean geometry in a finite number of dimensions that is different from elliptic and hyperbolic geometry (and from the Minkowskian geometry of spacetime). Here the linear structure is the same as the Euclidean one but distance is not "uniform" in all directions. Instead of the usual sphere in Euclidean space, the unit ball is a general symmetric convex set. Therefore, although the parallel axiom is valid, Pythagoras' theorem is not. This book begins by presenting the topological properties of Minkowski spaces, including the existence and essential uniqueness of Haar measure, followed by the fundamental metric properties - the group of isometries, the existence of certain bases and the existence of the Lowner ellipsoid. This is followed by characterizations of Euclidean space among normed spaces and a full treatment of two-dimensional spaces. The three central chapters present the theory of area and volume in normed spaces. The author describes the fascinating geometric interplay among the isoperimetrix (the convex body which solves the isoperimetric problem), the unit ball and their duals, and the ways in which various roles of the ball in Euclidean space are divided among them. The next chapter deals with trigonometry in Minkowski spaces and the last one takes a brief look at a number of numerical parameters associated with a normed space, including J. J. Schaffer's ideas on the intrinsic geometry of the unit sphere. Each chapter ends with a section of historical notes and the book ends with a list of 50 unsolved problems. Minkowski Geometry will appeal to students and researchers interested in geometry, convexity theory and functional analysis.


Язык: en

Рубрика: Математика/Геометрия и топология/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1996

Количество страниц: 346

Добавлена в каталог: 23.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abelian      37
Ader, O. B.      185
Affine      24 27 100 114 216 245 260
Affine invariant      54 95 96 122 137 139 140
Affine isometry      94
Affine transfoimation      178 185
Ailchison. P. W      97
Aleksandrov's theorem      196
Aleksandrov, A, D.      196 222 224
Alexander, R.      226
Algebra, multilinear      191
Alternating $k$-linear form      173 191 223
Amir, D      84 85 89 95 97
angle      251
Angle measure      122 251 265 268 269 311
Angle measure, axioms for      272
Annihilator      4
Antipodal point      130
Arc length      226 287
Arc length, Euclidean      119
Archimedes' theorem      311
Area      73 141
Area function, characterization of      183
Area of convex body      58 66
Area of sector      265
Area, continuity of      64 67
Area, equal in equal time      133 274
Area, intrinsic      182
Area, maximal      118
Area, minimal      138 144
Area, minimized by flat surfaces      141 223
Area, normalized      122
Area, problem of      145
Area, projected      73
Area, translation-invariant      118
Arithmetic-geometric mean inequality      82 232
Asplund, E      74 100 104 130
Associativity      6
Asymptotic properties      74
atom      191
Auerbach system      95
Auerbach, H.      95 96
Axiom of Choice      44
B - P property      see Busemann - Petty problem
Ball, K. M.      95 96 185 225 229 249
Bambah, R. P.      72
Banach - Mazur distance      63 73-5 85 96 138-9 196 234 276 277 282 300 309 310
Banach - Mazur distance, to cube      96
Banach - Mazur distance, to Euclidean space      75 98
Banach - Mazur metric      see Banach - Mazur distance
Banach space, flat      276 306
Banach space, local theory of      74 98
Banach space, reflexive      311 (see also reflexive)
Banach space, separable      280
Banach, S.      32 33 42 43 44 73 283
Barry, J. E.      273
Barthel, W      132 139 182 230 238 246 250 272
Basis      1
Basis of eigenvectors      176
Basis, dual      2 10 127 178
Basis, mutually normal      79
Basis, normal      261 262
Basis, orthonormal      80
Basis, usual, I basis constant      304
Behrend, F.      96 185
Benson, R. V.      118 132 140 156
Berger, M      71
Bertrand's theorem      86
Bertrand, M      86 97
Besicovilch, A. S.      142 250
Biberstein, O.      265 273
Birkhoff, G.      77 95 130
Blaschke - Santal$\acute{o}$ inequality      54 72 122 212 217 259
Blaschke addition      198 223 225 237
Blaschke addition, harmonic      199 201
Blaschke addition, radial      237
Blaschke difference body      199
Blaschke scalar product      198
Blaschke selection theorem      46 67 69 74 81 181
Blaschke sum      see Blaschke addition
Blaschke, W      54 69 72 74 77 91 97
Bliss, G. A.      272
Body, convex      see convex body
Body, intersection      see intersection body
Body, projection      see projection body
Bohnenblust, P.      89 131 278 304 305
Boju, V.      182
Bolker, E. D      187 190 222
Bollob$\Acute{a}$s, B      96
Bolts, T.      32 46 71
Boltyanski, V. G.      131 134
Bolzano. B.      9
Bonnesen's inequality      124 133 272
Bonnesen.T.      10 45 53 60 65 71 123 124 133 202 230 272
Borel $\sigma$ -algebra      53 135
Borel measure      40
Borel measure, regular      40 237
Borel measure, translation-invariant      36
Borel set      29 39 40 53 191 198
Bouligand, G      139 229 245
Bounded      14 29
Bounded, totally      29 67 68
Bourgain, J.      72 96 187 197 213 225
Brass, P.      130
Brothers, S. E      184
Brunn - Minkowski inequality      55 57 72 184 199 225 230 232 238
Brunn - Minkowski theorem      see Brunn - Minkowski inequality
Brunn - Minkowski theory      144
Brunn - Minkowski theory, generalizations of      183
Brunn's theorem      87 97
Brunn, H.      10 55 72 86 87 97
Burago, Yu. D.      73
Buscmann definition      139 150 212 229
Busemann - Petty problem      246-9
Busemann intersection inequality      214 244-5 250
Busemann intersection inequality, strengthening of      310
Busemann, H.      118 122 130 132 133 139 144 182 183 184 185 192 193 194 195 205 216 223 224 225 229 230 239 240 241 242 244 246 247 248 250 254 256 257 258 260 265 266 271 272 286 287
Bynum, W. L      43
Campi, S.      225
Carath$\acute{e}$odory condition      39 237
Carath$\acute{e}$odory's theorem      8
Carath$\acute{e}$odory, G.      10 39 77 95 130 250
Cardinality      210
Cartan, H.      44
Cartesian product      220 221
Cassels, J. W. S.      72
Castaing,C.      14
Cauchy - Schwarz inequality      20
Cauchy sequence      14 29 67
Cauchy's area formula      60 73 190 201 202 204
Cauchy's theorem      see Cauchy's area formula
Ceder, J. G.      130
Centre, false      97
Centre, metric      76
Centre, nine-point      100
Centre, pseudo      97
Centroid      105
Chakerian, G. D      95 108 109 131 132 133 205 222 243 244 304
Charzyrtski, Z.      95
Cheng, S. - Y.      224
Chilakamarri, K. B.      130
Chilton, B. L.      160 184
Choquet, G      139 197 229 245
Chords, mid-points of      86 87
Cieslik, D      134
Circle of Apollonius      126
Circle, nine-point      106 130
Circle, six-point      106 130
Circumcentre      100 106
circumcircle      104 105
Circumference      100
Circumradius      96 104-5 131
Clack,R      96 146 179 181 185 196
Cohn, B. L.      37 44
Combinatorial problems      129
Commutativity      6
Compact      14 67
Compact Hausdorff space, continuous functions on      280
Compact Hausdorff space, extremally disconnected      280 305
Compact set      29 61
Compact set, space of      73
Compactness of ($\mathcal{D}_{d}$, $\Delta$)      73
Compactness of ($\mathcal{K}_{a}$, $\delta$)      68
Compactness of unit ball      30 31
Compactness, sequential      46 67 69 74
Completeness      29 67
Completeness of ($\mathcal{K}$, $\delta$)      74
Component      193
Cone      192
Cone, double      129 170
Cone, double, intersection body of      154
Cone, double, intersection body of, dual of      154
Cone, double, over hexagon      167-9
Cone, double, over regular $n$-gon      184
Cone, double, projection body of dual      189
Cone, dual of      159
Cone, subtended      145
Connectedness of ($\mathcal{D}_{D}$, $\Delta$)      73
Constantin, R. F.      273
Continuous even functions      197
Contractive mapping      196 224 226 234 288 309 310
Convergence of sequence of sets      73
Convergence, absolute      283
Convergence, unconditional      283 284
Convex      6
Convex body      46 (see also convex set)
Convex body, boundary of      142
Convex body, sequence of      70
Convex body, smooth      144
Convex body, symmetric      55
Convex function      193
Convex hull      7 194
Convex set      1
Convex set, approximation of      46 74
Convex set, closed      46
Convex set, closed, with 0 an interior point      46
Convex set, compact      11
Convex set, compact, non-empty      46
Convex set, dimension of      8
Convex set, with analytic boundary      64 74
Convex set, with analytic boundary, density of      65
Convex sets, space of      74
Convex sets, space of, non-empty compact      69
Convex sets, space of, uniformly bounded      69
Convex, strictly      9 87
Convexity of $\sigma$      138 139 172
Convexity of $\sigma^{k}$      183
Convexity of unit ball      17
Convexity requirement      139
Convexity, degree of      304
Core      32
Cosine curve, solid of revolution of      160 184
Cosine function      252 256
Cosine transform      247
Cotangent space      173
Coxeter, H. S. M      160 184 188 222
Croft, H.T.      246
Cross-polytope      21 55 72 85 213 214 222 227 282
Cross-polytope, inscribed      80 140
Cross-polytope, inscribed, maximal      95
Cross-product      120
Cross-section      138 147 148 171 187 214 229 243 244 248 250 280
Cross-section and projection, duality between      72
Cross-section body      246-7
Cross-section function      246
Cross-section of intersection body      247
Cross-section of polytopes      246
Cross-section, $n$-dimensional      98
Cross-section, area of      151
Cross-section, congruent      250
Cross-section, dual of      151 188
Cross-section, homothetic      250
Cross-section, rhombic      148
Cross-section, two-dimensional      86 97
Crotty, J. M.      100 106 110 111 132
Crystallography      183-4 222
cube      9 55 96 138 141 148 151 170 171 176 198 214 219 222 227 247 249 289 293 298
Cube, circumscribed      96
Cube, circumscribed, unit      80
Cube, cross-sections of      249
Cube, dual of      158
Cube, high dimensional      222
Cube, intersection body of      153
Cube, intersection body of, dual of      153
Cube, projection body of dual      158
Cube, projection of      294
Cube, unit      20 27 70 170
Cubo-octahedron      148 170 215 227 300
Cubo-octahedron, affine regular      24
Cubo-octahedron, regular inscribed      104
Curvature      111 117 270
Curvature, Gaussian      10 224 249
Curvature, radius of      270
Curve      102 112 276 286 288 289 306
Curve of constant width      131
Curve, closed      287
Curve, convex      112
Curve, end point of      287
Curve, initial point of      287
Curve, length of      97 102 111 112 141 287
Curve, length of, minimizing      185
Curve, rectifiable      211 270 286 287
Curve, simple      287
Curve, simple, closed      112 291
Curve, simple, rectifiable      292
Cylinder      138 147 148 172 184 189
Cylinder, dual of      160
Cylinder, intersection body of      155
Cylinder, projection body of dual      160
Danzer, L      10 81
Davis, W. J.      305
Day, M. M      31 86 92 94 95 129 133
De Figueiredo, D. G.      92 97 133
Dean, D. W.      305
Derived geometry, first      263
Determinant      173
Diameter      85 96 131 237 238
Diameter of ($\mathcal{D}_{d}$, $\Delta$)      73
Diametrically maximal      130
Differentiability      29
Differential      175
Differential equation      271
Differential foims      173
Differential formulas      268
Differential geometry of Minkowski spaces      273
Differential operator      175 178
DIMENSION      1
Dimension of convex set      8
Dinghas, A.      184
Distance to Euclidean ball, maximal      96
Distance, Hausdorff      see Hausdorff metric
Distance, shortest      125
Distinct distances, minumum number of      130
Distributive law      6
Divergence      173 178 185
Divergence theorem      72 177 184 185
Djori$\acute{c}$, M      311
dodecahedron      166 167
Dodecahedron, rhombic      see rhombic dodecahedron
Double cone      see cone double
1 2 3 4
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