| Introduction: The Models | p. 1 |
| The Mathematical Models | p. 11 |
| The Monge-Kantorovich Problem | p. 11 |
| The Gilbert-Steiner Problem | p. 12 |
| Three Continuous Extensions of the Gilbert-Steiner Problem | p. 13 |
| Xia's Transport Paths | p. 13 |
| Maddalena-Solimini's Patterns | p. 14 |
| Traffic Plans | p. 14 |
| Questions and Answers | p. 16 |
| Plan | p. 17 |
| Related Problems and Models | p. 19 |
| Measures on Sets of Paths | p. 19 |
| Urban Transportation Models with more than One Transportation Means | p. 20 |
| Traffic Plans | p. 25 |
| Parameterized Traffic Plans | p. 27 |
| Stability Properties of Traffic Plans | p. 29 |
| Lower Semicontinuity of Length, Stopping Time, Averaged Length and Averaged Stopping Time | p. 30 |
| Multiplicity of a Traffic Plan and its Upper Semicontinuity | p. 31 |
| Sequential Compactness of Traffic Plans | p. 33 |
| Application to the Monge-Kantorovich Problem | p. 34 |
| Energy of a Traffic Plan and Existence of a Minimizer | p. 35 |
| The Structure of Optimal Traffic Plans | p. 39 |
| Speed Normalization | p. 39 |
| Loop-Free Traffic Plans | p. 41 |
| The Generalized Gilbert Energy | p. 42 |
| Rectifiability of Traffic Plans with Finite Energy | p. 44 |
| Appendix: Measurability Lemmas | p. 44 |
| Operations on Traffic Plans | p. 47 |
| Elementary Operations | p. 47 |
| Restriction, Domain of a Traffic Plan | p. 47 |
| Sum of Traffic Plans (or Union of their Parameterizations) | p. 48 |
| Mass Normalization | p. 48 |
| Concatenation | p. 48 |
| Concatenation of Two Traffic Plans | p. 48 |
| Hierarchical Concatenation (Construction of Infinite Irrigating Trees or Patterns) | p. 49 |
| A Priori Properties on Minimizers | p. 51 |
| An Assumption on [mu superscript +], [mu superscript -] and [pi] Avoiding Fibers with Zero Length | p. 51 |
| A Convex Hull Property | p. 53 |
| Traffic Plans and Distances between Measures | p. 55 |
| All Measures can be Irrigated for [alpha] > 1 - 1/N | p. 56 |
| Stability with Respect to [mu superscript +] and [mu superscript -] | p. 58 |
| Comparison of Distances between Measures | p. 59 |
| The Tree Structure of Optimal Traffic Plans and their Approximation | p. 65 |
| The Single Path Property | p. 65 |
| The Tree Property | p. 70 |
| Decomposition into Trees and Finite Graphs Approximation | p. 71 |
| Bi-Lipschitz Regularity | p. 77 |
| Interior and Boundary Regularity | p. 79 |
| Connected Components of a Traffic Plan | p. 79 |
| Cuts and Branching Points of a Traffic Plan | p. 81 |
| Interior Regularity | p. 82 |
| The Main Lemma | p. 82 |
| Interior Regularity when [characters not reproducible] | p. 85 |
| Interior Regularity when [mu superscript +] is a Finite Atomic Measure | p. 89 |
| Boundary Regularity | p. 91 |
| Further Regularity Properties | p. 93 |
| The Equivalence of Various Models | p. 95 |
| Irrigating Finite Atomic Measures (Gilbert-Steiner) and Traffic Plans | p. 95 |
| Patterns (Maddalena et al.) and Traffic Plans | p. 96 |
| Transport Paths (Qinglan Xia) and Traffic Plans | p. 97 |
| Optimal Transportation Networks as Flat Chains | p. 100 |
| Irrigability and Dimension | p. 105 |
| Several Concepts of Dimension of a Measure and Irrigability Results | p. 105 |
| Lower Bound on d([mu]) | p. 111 |
| Upper Bound on d([mu]) | p. 112 |
| Remarks and Examples | p. 114 |
| The Landscape of an Optimal Pattern | p. 119 |
| Introduction | p. 119 |
| Landscape Equilibrium and OCNs in Geophysics | p. 119 |
| A General Development Formula | p. 122 |
| Existence of the Landscape Function and Applications | p. 124 |
| Well-Definedness of the Landscape Function | p. 124 |
| Variational Applications | p. 127 |
| Properties of the Landscape Function | p. 128 |
| Semicontinuity | p. 128 |
| Maximal Slope in the Network Direction | p. 129 |
| Holder Continuity under Extra Assumptions | p. 131 |
| Campanato Spaces by Medians | p. 131 |
| Holder Continuity of the Landscape Function | p. 132 |
| The Gilbert-Steiner Problem | p. 135 |
| Optimum Irrigation from One Source to Two Sinks | p. 135 |
| Optimal Shape of a Traffic Plan with given Dyadic Topology | p. 143 |
| Topology of a Graph | p. 143 |
| A Recursive Construction of an Optimum with Full Steiner Topology | p. 144 |
| Number of Branches at a Bifurcation | p. 145 |
| Dirac to Lebesgue Segment: A Case Study | p. 151 |
| Analytical Results | p. 152 |
| The Case of a Source Aligned with the Segment | p. 152 |
| A "T Structure" is not Optimal | p. 153 |
| The Boundary Behavior of an Optimal Solution | p. 155 |
| Can Fibers Move along the Segment in the Optimal Structure? | p. 159 |
| Numerical Results | p. 159 |
| Coding of the Topology | p. 159 |
| Exhaustive Search | p. 160 |
| Heuristics for Topology Optimization | p. 160 |
| Multiscale Method | p. 161 |
| Optimality of Subtrees | p. 164 |
| Perturbation of the Topology | p. 165 |
| Application: Embedded Irrigation Networks | p. 169 |
| Irrigation Networks made of Tubes | p. 169 |
| Anticipating some Conclusions | p. 171 |
| Getting Back to the Gilbert Functional | p. 172 |
| A Consequence of the Space-filling Condition | p. 175 |
| Source to Volume Transfer Energy | p. 176 |
| Final Remarks | p. 177 |
| Open Problems | p. 179 |
| Stability | p. 179 |
| Regularity | p. 179 |
| The who goes where Problem | p. 180 |
| Dirac to Lebesgue Segment | p. 180 |
| Algorithm or Construction of Local Optima | p. 181 |
| Structure | p. 182 |
| Scaling Laws | p. 183 |
| Local Optimality in the Case of Non Irrigability | p. 183 |
| Skorokhod Theorem | p. 185 |
| Flows in Tubes | p. 189 |
| Poiseuille's Law | p. 189 |
| Optimality of the Circular Section | p. 190 |
| Notations | p. 191 |
| References | p. 193 |
| Index | p. 199 |
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