Mass Transportation Problems, Volume I, Table of Contents
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Contents to Volume I | |||||
Preface to Volume I | vii | ||||
Preface to Volume II | xv | ||||
1 | Introduction | 1 | |||
1.1 | Mass Transportation Problems in Probability Theory | 1 | |||
1.2 | Specially Structured Uansportation Problems | 21 | |||
1.3 | Two Examples of the Interplay Between Continuous and Discrete MTPs | 23 | |||
1.4 | Stochastic Applications | 27 | |||
2 | The Monge-Kantorovich Problem | 57 | |||
2.1 | The Multivariate Monge-Kantorovich Problem: An Introduction | 58 | |||
2.2 | Primal and Dual Monge-Kantorovich Functionals | 64 | |||
2.3 | Duality Theorems in a Topological Setting | 76 | |||
2.4 | General Duality Theorem | 82 | |||
2.5 | Duality Theorems with Metric Cost Functions | 86 | |||
2.6 | Dual Representation for Lp-Minimal Metrics | 96 | |||
3 | Explicit Results for the Monge-Kantorovich Problem | 107 | |||
3.1 | The One-Dimensional Case | 107 | |||
3.2 | The Convex Case | 112 | |||
3.3 | The General Case | 123 | |||
3.4 | An Extension of the Kantorovich L2-Minimal Problem | 132 | |||
3.5 | Maximum Probability of Sets, Maximum of Sums, and Stochastic Order | 144 | |||
3.6 | Hoeffding-Fréchet Bounds | 151 | |||
3.7 | Bounds for the Total lyansportation Cost | 158 | |||
4 | Duality Theory for Mass Transfer Problems | 161 | |||
4.1 | Duality in the Compact Case | 161 | |||
4.2 | Cost Functions with Triangle Inequality | 172 | |||
4.3 | Reduction Theorems | 190 | |||
4.4 | Proofs of the Main Duality Theorenis and a Discussion | 207 | |||
4.5 | Duality Theorems for Noncompact Spaces | 219 | |||
4.6 | Infinite Linear Programs | 241 | |||
4.6.1 | Duality Theory for an Abstract Scheme of Infinite-Dimensional Linear Programs and Its Application to the Mass Transfer Problem | 241 | |||
4.6.2 | Duality Theorems for the Mass Transfer Problem with Given Marginals | 245 | |||
4.6.3 | Duality Theorem for a Marginal Problem with Additional Constraints of Moment-Type | 251 | |||
4.6.4 | Duality theorem for a Further Extremal Marginal Problem | 258 | |||
4.6.5 | Duality Theorem for a Nontopological Version of the Mass Transfer Problem | 265 | |||
5 | Applications of the Duality Theory | 275 | |||
5.1 | Mass Transfer Problem with a Smooth Cost Function-Explicit Solution | 275 | |||
5.2 | Extension and Approximate Extension Theorems | 290 | |||
5.2.1 | The Simplest Extension Theorem (the Case X = E(S) and X1 = E(S1)) | 290 | |||
5.2.2 | Approximate Extension Theorems | 292 | |||
5.2.3 | Extension Theorenis | 295 | |||
5.2.4 | A continuous selection theorem | 302 | |||
5.3 | Approximation Theorems | 306 | |||
5.4 | An Application of the Duality Theory to the Strassen Theorem | 319 | |||
5.5 | Closed Preorders and Continuous Utility Functions | 322 | |||
5.5.1 | Statement of the Problem and the Idea of the Duality Approach | 322 | |||
5.5.2 | Functionally Closed Preorders | 324 | |||
5.5.3 | Two Generalizations of the Debreu Theorem | 329 | |||
5.5.4 | The Case of a Locally Compact Space | 335 | |||
5.5.5 | Varying preorders and a universal utility theorem | 337 | |||
5.5.6 | Functionally Closed Preorders and Strong Stochastic Dominance | 341 | |||
5.6 | Further Applications to Utility Theory | 344 | |||
5.6.1 | Preferences That Admit Lipschitz or Continuous Utility Functions | 344 | |||
5.6.2 | Application to Choice Theory in Mathematical Economics | 352 | |||
5.7 | Applications to Set-Valued Dynamical Systems | 354 | |||
5.7.1 | Compact-Valued Dynamical Systems: Quasiperiodic Points | 354 | |||
5.7.2 | Compact-Valued Dynamical Systems: Asymptotic Behavior of Trajectories | 358 | |||
5.7.3 | A Dynamic Optimization Problem | 363 | |||
5.8 | Compensatory Transfers and Action Profiles | 367 | |||
6 | Mass Transshipment Problems and Ideal Metrics | 371 | |||
6.1 | Kantorovich-Rubinstein Problems with Constraints | 372 | |||
6.2 | Constraints on the k-th Difference of Marginals | 383 | |||
6.3 | The General Case | 402 | |||
6.4 | Minimality of Ideal Metrics | 414 | |||
References | 429 | ||||
Abbreviations | 473 | ||||
Symbols | 475 | ||||
Index | 478 |