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Webster R. — Convexity
Webster R. — Convexity

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Название: Convexity

Автор: Webster R.

Аннотация:

This text provides a comprehensive introduction to convex sets and functions. Requiring only a basic knowledge of analysis and linear algebra, the book clearly discusses topics as diverse as number theory, classical extremum problems, combinatorial geometry, linear programming, game theory, polytopes, bodies of constant width, the gamma function, minimax approximation, and the theory of linear, classical, and matrix inequalities. The book clearly shows how convexity links many different topics in mathematics, from linear algebra to analysis. Suitable for upper-level undergraduate and graduate students, this book offers full solutions to over 200 exercises as well as detailed suggestions for further reading.


Язык: en

Рубрика: Математика/Геометрия и топология/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1994

Количество страниц: 464

Добавлена в каталог: 18.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Acute angle      27
Addition of sets      2
Addition of vectors      2
Affine basis      15
Affine combination      7
Affine dependence      135
Affine hull      7
Affine manifold      5
Affine set      5
Affine transformation      22
Affine transformation, continuity of      44 45
Affine transformation, non-singular      24
Affine transformation, satisfies Lipschitz condition      44
Affine variety      5
Affinely dependent set      10
Affinely independent set      10
Angle between vectors      27
Apex of a pyramid      116
Archimedes      262
Arithmetic mean      201
Artificial variable      177
Artin’s characterization of $\Gamma$      209 211
Associative law      331 332 333 337
Augmented problem      176
Axiomatic convexity      330
Ball      32
Ball, volume      278 279
Barbier’s theorem      376
Barycentric coordinates      15
Base of a bipyramid      117
Base of a prism      119
Base of a pyramid      116
Basic optimal vector      165
Basic solution      163 164
Basic variable      172
Best approximation of degree n      324 327
beta function      206 215
Bipyramid      117
Birkhoff, G.      245
Birkhoffs theorem      245 246 247 251
Blaschke selection principle      90 97 303
Blumenthal, L. M.      320
Boundary      36
Boundary point      36
Bounded sequence      39
Bounded set      40
Brouwer’s fixed-point theorem      356 361 362
Brunn — Minkowski theorem      297 301 303
Brunn, H.      297 336
Brunn’s inequality      297 299 301 307 314
Brunn’s theorem      337
Canonical problem      165
CAP      313
Caratheodory’s theorem      55 60
Cauchy — Schwarz inequality      26
cell      254
Centre of a ball      32
Centre of a convex body      319
Chebyshev polynomial      330
Chebyshev set      362
Chord of a convex body      319
Circular arc      373
Circumball      95
Circumcentre      95
Circumradius      94
Classical inequalities      200
Closed ball      32
Closed convex hull      63
Closed disc      32
Closed halfflat      84
Closed halfspace      20
Closed set      35
Closed unit ball      32
Closure      34
Closure point      34
Coface      135
Collinearity-preserving mapping      343
Column player      181
Combinatorial equivalence      108
Combinatorial geometry      127 316 318 320
Commutative law      332 334 342
Compact set      40
Complementary slackness theorem      161 187
Complete convex body      371
Concave function      193 217
Cone      75
Cone construction      116
Congruence transformation      29
Congruent copy      30
Congruent sets      30
Conical set      276
Conical set, volume      276 277
Consequence of linear inequalities      154
Consistent inequalities      150 154
Constant width      367 370
Continuity      43 195 196 224
Continuity of      195 196 224 227—230
Continuity of mixed volumes      289
Continuity of set functions      97
Continuity of, differentiability of      194—197 199 224—230
Continuity of, support of      197 220
Convergence of points      38
Convergence of sets      93
Convex body      94
Convex body of constant width      367 370
Convex body, complete      371
Convex combination      51
Convex cone generated by a set      76
Convex function      193 217
Convex hull      54 85 342
Convex polytope      105
Convex programming problem      239
Convex set      49 335
Convexity-preserving mapping      343
Coordinate      2
Core      337
Core, point      337
Cross multiplication      332
Cyclic polytope      120 121
Cycling      179
Cylindrical set      276
Cylindrical set, volume      276 277
Dehn — Sommerville equations      132 133
Derivative      227
Determination of a face      107
Diagonal      250
Diagonal leading      250
Diagonal positive      260
Diameter      58
Difference body      368
Differentiability      194—197 199 224—230
Differentiate      227
DIMENSION      10 14
Direction of a halfline      74
Direction of a line      77
Direction of linearity      77
Direction of recession      76
Directional derivative      225
Directional derivative, one-sided      226
Distance between points      26
Distance between sets      91
Distance function      45 218
Dominated strategy      190
Double cone construction      117
Double-polar dual      101
Doubly stochastic matrix      245 246
Dual pair      152
Dual problem      160
Duality      99
Duality theorem      157 161 181 185
EDGE      107
Elementary matrix      268
Elementary set      254 255
Elementary vector      2
Elements      1 105 330
Ellipsoid      27
Ellipsoid, volume      279
Epigraph      218
Equilibrium theorem      161
Equioscillatlon properties      327
Error of polynomial approximation      324 326
Essential strategy      187
Euclid      1 105 330 346
Euclidean space      1 2
Euler hyperplane      134
Euler — Poincare relation      127
Euler, L.      127 217 367
Euler’s constant      216
Euler’s relation      115 116 117 118 119 124 127
Existence laws      332
Expected gain, loss      182 183
Exposed face      86 234
Exposed point      88
Extension      331 332 333
Extreme direction      80
Extreme halfline      80
Extreme point      80
Face      79
Face, exposed      86 234
Face, improper      80
Face, proper      80
Facet      110
Farey sequence      351
Farkas lemma      156
Farthest point      88 365
Farthest-point mapping      365
Feasible problem      157 160
Feasible set      157 160 165 239
Feasible vector      157 160 165
Final tableau      173 175
Finite basis for a polyhedral set      147
Finite basis theorem      145 146 147
Finite distance apart      91
Finite generation of a polyhedral set      147
Finitely generated convex cone      145
Fixed point      356 366
Fixed-point property      366
Flat      4 5 342
Frobenius, G.      362
G-sequence      142
Gale diagram      144
Gale transform      135 137
Gale’s evenness condition      123
Game theory      181
Gamma function      206 208
Gauge function      236
Gauss — Lucas Theorem      59
Gauss, C. F.      209
Geometric mean      201
Geometry of numbers      350 351
Good approximation      352
Gradient      227
Graph      218
Hadamard, J.      249
Hadamard’s inequality      249
Hadwiger, H.      297
Halfflat      84
Halfline      74
Halfplane      20
Halfspace      19 20
Harmonic mean      203 204
Hausdorff distance (metric)      92
Helly, E.      316
Helly’s theorem      316 319
Hessian matrix      230
Holder’s Inequality      202 207
Homothetic sets      4 297
Homothety      362
Hyperplane      17
Idempotent laws      332
Improper face      80
Inball      95
Incentre      95
Inconsistent inequalities      150 154
Infeasible problem      157 160
Infinite product for sinx      211
Initial point of a halfline      74
Initial tableau      172 175
Inner product      25
Inner-volume      262
Inradius      94
Insoluble problem      157 160
Interior      33
Interior point      33
Isodiametric inequality      303 307 312 314
Isodiametric problem      98 303
Isoperimetric inequality      303 307 314
Isoperimetric problem      297 303 304 306
ith pure strategy      182
Jensen’s Inequality      200 206 217
Join of points      330 333
Join of sets      331 333
Join space      330 333
jth pure strategy      182
Jung, H. W. E.      320
Jung’s theorem      320
k-face      80
k-neighbourly      120 125
Kantorovich’s inequality      253
Kay, D. C.      343
Kernel      336
Kirchberger, P.      318
Kirchberger’s theorem      318
Krasnosselsky, M. A.      323
Krasnosselsky’s theorem      323
Krein — Milman theorem      86
Kuhn — Tucker conditions      239 242
Lagrange’s four square theorem      354
Lagrange’s interpolation formula      324
Lagrangian function      240
Lattice point      2
Leading diagonal      250
Legal linear combination      154
Legendre’s duplication formula      213
Length of a cell in $\mathbb{R}^1$      257
Length of a vector      26
Level set      222
Line      5
Line segment      5 49
Lineality space      78
Linear access      337 339
Linear access, point      339
linear combination      8
Linear hull      8
Linear in a direction      77
Linear inequalities      149
Linear manifold      5
Linear programming      156
Linear variety      5
Lipschitz condition      44
log-convexity      206 207
Lower integral      275
Lower sum      275
Lyusternik, L. A.      297
Matrix game      181
Matrix inequalities      244
Maximal chord length      367 369
Maximal width      368
Mean of order t      204
Meyer, W.      343
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