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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)
DOI: https://doi.org/10.1090/memo/1180
Published electronically: June 13, 2017
Keywords: Ricci curvature,
Monge-Kantorovich problem,
needle decomposition,
Riemannian geometry
MSC: Primary 53C21, 52A20, 52A40
Table of Contents
Chapters
- 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
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.- Luigi Ambrosio, Lecture notes on optimal transport problems, Mathematical aspects of evolving interfaces (Funchal, 2000) Lecture Notes in Math., vol. 1812, Springer, Berlin, 2003, pp. 1–52. MR 2011032, DOI 10.1007/978-3-540-39189-0_{1}
- D. Bakry and Michel Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177–206 (French). MR 889476, DOI 10.1007/BFb0075847
- Dominique Bakry and Zhongmin Qian, Some new results on eigenvectors via dimension, diameter, and Ricci curvature, Adv. Math. 155 (2000), no. 1, 98–153. MR 1789850, DOI 10.1006/aima.2000.1932
- Dominique Bakry, Ivan Gentil, and Michel Ledoux, Analysis and geometry of Markov diffusion operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 348, Springer, Cham, 2014. MR 3155209
- Bayle, V., Propriétés de Concavité du Profil Isopérimétrique et Applications. Ph.D. thesis, Institut Joseph Fourier, Grenoble, 2004. Available at \verb"http://www.youscribe.com"
- S. Bobkov, Extremal properties of half-spaces for log-concave distributions, Ann. Probab. 24 (1996), no. 1, 35–48. MR 1387625, DOI 10.1214/aop/1042644706
- S. G. Bobkov, On concentration of distributions of random weighted sums, Ann. Probab. 31 (2003), no. 1, 195–215. MR 1959791, DOI 10.1214/aop/1046294309
- S. G. Bobkov and F. L. Nazarov, Sharp dilation-type inequalities with fixed parameter of convexity, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 351 (2007), no. Veroyatnost′i Statistika. 12, 54–78, 299 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 152 (2008), no. 6, 826–839. MR 2742901, DOI 10.1007/s10958-008-9100-9
- C. Borell, Convex set functions in $d$-space, Period. Math. Hungar. 6 (1975), no. 2, 111–136. MR 404559, DOI 10.1007/BF02018814
- Herm Jan Brascamp and Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), no. 4, 366–389. MR 0450480, DOI 10.1016/0022-1236(76)90004-5
- Keith Burns and Marian Gidea, Differential geometry and topology, Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2005. With a view to dynamical systems. MR 2148643
- Peter Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213–230. MR 683635
- Luis A. Caffarelli, Mikhail Feldman, and Robert J. McCann, Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs, J. Amer. Math. Soc. 15 (2002), no. 1, 1–26. MR 1862796, DOI 10.1090/S0894-0347-01-00376-9
- Cayley, A., On Monge’s “Mémoire sur la Théorie des Déblais et des Remblais.” Proc. London Math. Soc., Vol. s1-14, Issue 1, (1882), 139–143. Available at \verb"http://dx.doi.org/10.1112/plms/s1-14.1.139"
- Isaac Chavel, Riemannian geometry, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 98, Cambridge University Press, Cambridge, 2006. A modern introduction. MR 2229062
- Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, AMS Chelsea Publishing, Providence, RI, 2008. Revised reprint of the 1975 original. MR 2394158
- Dario Cordero-Erausquin, Robert J. McCann, and Michael Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math. 146 (2001), no. 2, 219–257. MR 1865396, DOI 10.1007/s002220100160
- Dario Cordero-Erausquin, Robert J. McCann, and Michael Schmuckenschläger, Prékopa-Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport, Ann. Fac. Sci. Toulouse Math. (6) 15 (2006), no. 4, 613–635 (English, with English and French summaries). MR 2295207
- C. H. Edwards Jr., Advanced calculus of several variables, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1973 original. MR 1319337
- L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc. 137 (1999), no. 653, viii+66. MR 1464149, DOI 10.1090/memo/0653
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
- Mikhail Feldman and Robert J. McCann, Monge’s transport problem on a Riemannian manifold, Trans. Amer. Math. Soc. 354 (2002), no. 4, 1667–1697. MR 1873023, DOI 10.1090/S0002-9947-01-02930-0
- Matthieu Fradelizi, Concentration inequalities for $s$-concave measures of dilations of Borel sets and applications, Electron. J. Probab. 14 (2009), no. 71, 2068–2090. MR 2550293, DOI 10.1214/EJP.v14-695
- Matthieu Fradelizi and Olivier Guédon, The extreme points of subsets of $s$-concave probabilities and a geometric localization theorem, Discrete Comput. Geom. 31 (2004), no. 2, 327–335. MR 2060645, DOI 10.1007/s00454-003-2868-y
- Wilfrid Gangbo, The Monge mass transfer problem and its applications, Monge Ampère equation: applications to geometry and optimization (Deerfield Beach, FL, 1997) Contemp. Math., vol. 226, Amer. Math. Soc., Providence, RI, 1999, pp. 79–104. MR 1660743, DOI 10.1090/conm/226/03236
- Gromov, M., Paul Levy’s isoperimetric inequality. Appendix C in the book Metric structures for Riemannian and non-Riemannian spaces by M. Gromov. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2007.
- Gromov, M., Isoperimetric inequalities in Riemannian manifolds. Appendix I in the book Asymptotic Theory of Finite Dimensional Normed Spaces by V. D. Milman and G. Schechtman. Lecture notes in Math., Vol. 1200, Springer-Verlag, Berlin, 1986.
- M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003), no. 1, 178–215. MR 1978494, DOI 10.1007/s000390300004
- M. Gromov and V. D. Milman, Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces, Compositio Math. 62 (1987), no. 3, 263–282. MR 901393
- Ernst Heintze and Hermann Karcher, A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 451–470. MR 533065
- R. Kannan, L. Lovász, and M. Simonovits, Isoperimetric problems for convex bodies and a localization lemma, Discrete Comput. Geom. 13 (1995), no. 3-4, 541–559. MR 1318794, DOI 10.1007/BF02574061
- L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. MR 664597
- Michel Ledoux, Spectral gap, logarithmic Sobolev constant, and geometric bounds, Surveys in differential geometry. Vol. IX, Surv. Differ. Geom., vol. 9, Int. Press, Somerville, MA, 2004, pp. 219–240. MR 2195409, DOI 10.4310/SDG.2004.v9.n1.a6
- Peter Li and Shing Tung Yau, Estimates of eigenvalues of a compact Riemannian manifold, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 205–239. MR 573435
- László Lovász, Hit-and-run mixes fast, Math. Program. 86 (1999), no. 3, Ser. A, 443–461. MR 1733749, DOI 10.1007/s101070050099
- L. Lovász and M. Simonovits, Random walks in a convex body and an improved volume algorithm, Random Structures Algorithms 4 (1993), no. 4, 359–412. MR 1238906, DOI 10.1002/rsa.3240040402
- Emanuel Milman, Sharp isoperimetric inequalities and model spaces for the curvature-dimension-diameter condition, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 5, 1041–1078. MR 3346688, DOI 10.4171/JEMS/526
- Emanuel Milman, On the role of convexity in isoperimetry, spectral gap and concentration, Invent. Math. 177 (2009), no. 1, 1–43. MR 2507637, DOI 10.1007/s00222-009-0175-9
- Frank Morgan, Manifolds with density, Notices Amer. Math. Soc. 52 (2005), no. 8, 853–858. MR 2161354
- F. Nazarov, M. Sodin, and A. Vol′berg, The geometric Kannan-Lovász-Simonovits lemma, dimension-free estimates for the distribution of the values of polynomials, and the distribution of the zeros of random analytic functions, Algebra i Analiz 14 (2002), no. 2, 214–234 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 14 (2003), no. 2, 351–366. MR 1925887
- L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), 286–292 (1960). MR 117419, DOI 10.1007/BF00252910
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Elias M. Stein and Rami Shakarchi, Real analysis, Princeton Lectures in Analysis, vol. 3, Princeton University Press, Princeton, NJ, 2005. Measure theory, integration, and Hilbert spaces. MR 2129625
- Neil S. Trudinger and Xu-Jia Wang, On the Monge mass transfer problem, Calc. Var. Partial Differential Equations 13 (2001), no. 1, 19–31. MR 1854255, DOI 10.1007/PL00009922
- Hassler Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89. MR 1501735, DOI 10.1090/S0002-9947-1934-1501735-3
- DaGang Yang, Lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature, Pacific J. Math. 190 (1999), no. 2, 383–398. MR 1722898, DOI 10.2140/pjm.1999.190.383
- Jia Qing Zhong and Hong Cang Yang, On the estimate of the first eigenvalue of a compact Riemannian manifold, Sci. Sinica Ser. A 27 (1984), no. 12, 1265–1273. MR 794292