Skip to Main Content

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

Needle Decompositions in Riemannian Geometry

About this Title

Bo’az Klartag, Department of Mathematics, Weizmann Institute of Science, Rehovot 76100 Israel, and School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978 Israel

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)
Published electronically: June 13, 2017
Keywords: Ricci curvature, Monge-Kantorovich problem, needle decomposition, Riemannian geometry
MSC: Primary 53C21, 52A20, 52A40

View full volume PDF

View other years and numbers:

Table of Contents


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


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.

References [Enhancements On Off] (What's this?)