Mass Transportation Problems: Volume I: Theory (Probability and its Applications)

Contents to Volume I......Page 20
Preface to Volume I......Page 8
Preface to Volume II......Page 16
1.1 Mass Transportation Problems in Probability Theory......Page 28
1.2 Specially Structured Transportation Problems......Page 48
1.3 Two Examples of the Interplay Between Continuous and Discrete MTPs......Page 50
1.4 Stochastic Applications......Page 54
2 The Monge–Kantorovich Problem......Page 84
2.1 The Multivariate Monge–Kantorovich Problem: An Introduction......Page 85
2.2 Primal and Dual Monge–Kantorovich Functionals......Page 91
2.3 Duality Theorems in a Topological Setting......Page 103
2.4 General Duality Theorem......Page 109
2.5 Duality Theorems with Metric Cost Functions......Page 113
2.6 Dual Representation for L[sub(p)]-Minimal Metrics......Page 123
3.1 The One-Dimensional Case......Page 134
3.2 The Convex Case......Page 139
3.3 The General Case......Page 150
3.4 An Extension of the Kantorovich L[sub(2)]-Minimal Problem......Page 159
3.5 Maximum Probability of Sets, Maximum of Sums, and Stochastic Order......Page 171
3.6 Hoeffding–Fréchet Bounds......Page 178
3.7 Bounds for the Total Transportation Cost......Page 185
4.1 Duality in the Compact Case......Page 188
4.2 Cost Functions with Triangle Inequality......Page 199
4.3 Reduction Theorems......Page 217
4.4 Proofs of the Main Duality Theorems and a Discussion......Page 234
4.5 Duality Theorems for Noncompact Spaces......Page 246
4.6.1 Duality Theory for an Abstract Scheme of Infinite-Dimensional Linear Programs and Its Application to the Mass Transfer Problem......Page 268
4.6.2 Duality Theorems for the Mass Transfer Problem with Given Marginals......Page 272
4.6.3 Duality Theorem for a Marginal Problem with Additional Constraints of Moment-Type......Page 278
4.6.4 Duality Theorem for a Further Extremal Marginal Problem......Page 285
4.6.5 Duality Theorem for a Nontopological Version of the Mass Transfer Problem......Page 292
5.1 Mass Transfer Problem with a Smooth Cost Function—Explicit Solution......Page 302
5.2.1 The Simplest Extension Theorem (the Case X = E(S) and X[sub(1)] = E(S[sub(1)]))......Page 317
5.2.2 Approximate Extension Theorems......Page 319
5.2.3 Extension Theorems......Page 322
5.2.4 A continuous selection theorem......Page 329
5.3 Approximation Theorems......Page 333
5.4 An Application of the Duality Theory to the Strassen Theorem......Page 346
5.5.1 Statement of the Problem and the Idea of the Duality Approach......Page 349
5.5.2 Functionally Closed Preorders......Page 351
5.5.3 Two Generalizations of the Debreu Theorem......Page 356
5.5.4 The Case of a Locally Compact Space......Page 362
5.5.5 Varying preorders and a universal utility theorem......Page 364
5.5.6 Functionally Closed Preorders and Strong Stochastic Dominance......Page 368
5.6.1 Preferences That Admit Lipschitz or Continuous Utility Functions......Page 371
5.6.2 Application to Choice Theory in Mathematical Economics......Page 379
5.7.1 Compact-Valued Dynamical Systems: Quasiperiodic Points......Page 381
5.7.2 Compact-Valued Dynamical Systems: Asymptotic Behavior of Trajectories......Page 385
5.7.3 A Dynamic Optimization Problem......Page 390
5.8 Compensatory Transfers and Action Profiles......Page 394
6 Mass Transshipment Problems and Ideal Metrics......Page 398
6.1 Kantorovich–Rubinstein Problems with Constraints......Page 399
6.2 Constraints on the κth Difference of Marginals......Page 410
6.3 The General Case......Page 429
6.4 Minimality of Ideal Metrics......Page 441
References......Page 456
Abbreviations......Page 499
Symbols......Page 502
A......Page 516
B......Page 517
C......Page 518
D......Page 520
E......Page 522
G......Page 523
H......Page 524
K......Page 525
L......Page 526
M......Page 528
N......Page 531
P......Page 532
R......Page 534
S......Page 535
T......Page 537
W......Page 539
Z......Page 540