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Title page

Chapter 1. Introduction

Chapter 2. Regularity of geodesic foliations

2.1. Transport rays

2.2. Whitney’s extension theorem for 𝐶^{1,1}

2.3. Riemann normal coordinates

2.4. Proof of the regularity theorem

Chapter 3. Conditioning a measure with respect to a geodesic foliation

3.1. Geodesics emanating from a 𝐶^{1,1}-hypersurface

3.2. Decomposition into ray clusters

3.3. Needles and Ricci curvature

Chapter 4. The Monge-Kantorovich problem

Chapter 5. Some applications

5.1. The inequalities of Buser, Ledoux and E. Milman

5.2. A Poincaré inequality for geodesically-convex domains

5.3. The isoperimetric inequality and its relatives

Chapter 6. Further research

Appendix: The Feldman-McCann proof of Lemma 2.4.1

Bibliography

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