Needle Decompositions in Riemannian Geometry.
Klartag, Bo'az.
Needle Decompositions in Riemannian Geometry. - 1st ed. - 1 online resource (90 pages) - Memoirs of the American Mathematical Society ; v.249 . - Memoirs of the American Mathematical Society .
Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Regularity of geodesic foliations -- 2.1. Transport rays -- 2.2. Whitney's extension theorem for ^ -- 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 ^-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 -- Back Cover.
The localization technique from convex geometry is generalized to the setting of Riemannian manifolds whose Ricci curvature is bounded from below. In a nutshell, the author's 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 the author's analysis.
9781470441272
Curvature.
Decomposition (Mathematics).
Geometry, Riemannian.
Electronic books.
QA645 .K537 2017
516.362
Needle Decompositions in Riemannian Geometry. - 1st ed. - 1 online resource (90 pages) - Memoirs of the American Mathematical Society ; v.249 . - Memoirs of the American Mathematical Society .
Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Regularity of geodesic foliations -- 2.1. Transport rays -- 2.2. Whitney's extension theorem for ^ -- 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 ^-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 -- Back Cover.
The localization technique from convex geometry is generalized to the setting of Riemannian manifolds whose Ricci curvature is bounded from below. In a nutshell, the author's 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 the author's analysis.
9781470441272
Curvature.
Decomposition (Mathematics).
Geometry, Riemannian.
Electronic books.
QA645 .K537 2017
516.362