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Needle Decompositions in Riemannian Geometry


About this Title

Bo’az Klartag

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 249, Number 1180
ISBNs: 978-1-4704-2542-5 (print); 978-1-4704-4127-2 (online)
DOI: https://doi.org/10.1090/memo/1180
Published electronically: June 13, 2017
Keywords:Ricci curvature, Monge-Kantorovich problem, needle decomposition, Riemannian geometry

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Regularity of geodesic foliations
  • Chapter 3. Conditioning a measure with respect to a geodesic foliation
  • Chapter 4. The Monge-Kantorovich problem
  • Chapter 5. Some applications
  • Chapter 6. Further research
  • Appendix: The Feldman-McCann proof of Lemma 2.4.1

Abstract


The localization technique from convex geometry is generalized to the setting of Riemannian manifolds whose Ricci curvature is bounded from below. In a nutshell, our method is based on the following observation: When the Ricci curvature is non-negative, log-concave measures are obtained when conditioning the Riemannian volume measure with respect to a geodesic foliation that is orthogonal to the level sets of a Lipschitz function. The Monge mass transfer problem plays an important role in our analysis.

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