Poincaré Inequalities and Moment Maps
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 1, p. 1-41

We discuss a method for obtaining Poincaré-type inequalities on arbitrary convex bodies in n . Our technique involves a dual version of Bochner’s formula and a certain moment map, and it also applies to some non-convex sets. In particular, we generalize the central limit theorem for convex bodies to a class of non-convex domains, including the unit balls of p -spaces in n for 0<p<1.

Nous explorons un procédé de preuve d’inégalités de type Poincaré sur les corps convexes de n . Notre technique utilise une version duale de la formule de Bochner et une application moment. Elle s’applique également à certains corps non-convexes. En particulier, nous généralisons le théorème central limite pour les ensembles convexes à une classe de domaines non-convexes, qui comprend les boules unités de n munies de la norme p pour 0<p<1.

@article{AFST_2013_6_22_1_1_0,
     author = {Klartag, Bo'az},
     title = {Poincar\'e Inequalities and Moment Maps},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 22},
     number = {1},
     year = {2013},
     pages = {1-41},
     doi = {10.5802/afst.1366},
     zbl = {1279.60036},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2013_6_22_1_1_0}
}
Klartag, Bo’az. Poincaré Inequalities and Moment Maps. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 1, pp. 1-41. doi : 10.5802/afst.1366. http://www.numdam.org/item/AFST_2013_6_22_1_1_0/

[1] Abreu (M.).— Kähler geometry of toric manifolds in symplectic coordinates. Symplectic and contact topology: interactions and perspectives. Fields Inst. Commun., 35, Amer. Math. Soc., Providence, RI, p. 1-24 (2003). | MR 1969265 | Zbl 1044.53051

[2] Abreu (M.).— Kähler metrics on toric orbifolds. J. Differential Geom., 58, no. 1, p. 151-187 (2001). | MR 1895351 | Zbl 1035.53055

[3] Anttila (M.), Ball (K.), Perissinaki (I.).— The central limit problem for convex bodies. Trans. Amer. Math. Soc., 355, no. 12, p. 4723-4735 (2003). | MR 1997580 | Zbl 1033.52003

[4] Avkhadiev (F.), Wirths (K.-J.).— Unified Poincaré and Hardy inequalities with sharp constants for convex domains. ZAMM Z. Angew. Math. Mech. 87, no. 8-9, p. 632-642 (2007). | MR 2354734 | Zbl 1145.26005

[5] Bakry (D.), Émery (M.).— Diffusions hypercontractives (French). Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., 1123, Springer, Berlin, p. 177-206 (1985). | Numdam | MR 889476 | Zbl 0561.60080

[6] Barthe (F.), Cordero-Erausquin (D.).— Invariances in variance estimates, Proc. London Math. Soc. 106, (2013) p. 33-64. | Zbl 1281.60020

[7] Bobkov (S. G.).— On concentration of distributions of random weighted sums. Ann. Prob., 31, no. 1, p. 195-215 (2003). | MR 1959791 | Zbl 1015.60019

[8] Brascamp (H. J.), Lieb (E. H.).— 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. Funct. Anal., 22, no. 4, p. 366-389 (1976). | MR 450480 | Zbl 0334.26009

[9] Brezis (H.).— Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. | MR 2759829 | Zbl 1220.46002

[10] Brezis (H.), Marcus (M.).— Hardy’s inequalities revisited. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Vol. 25, p. 217-237 (1997). | Numdam | MR 1655516 | Zbl 1011.46027

[11] Cannas da Silva (A.).— Lectures on Symplectic Geometry. Lecture Notes in Math., 1764, Springer-Verlag (2008). | Zbl 1016.53001

[12] Chiang (Y.-J.).— Harmonic Maps of V-Manifolds. Ann. Global Anal. Geom., Vol. 8, No. 3, p. 315-344 (1990). | MR 1089240 | Zbl 0679.58014

[13] Diaconis (P.), Freedman (D.).— Asymptotics of graphical projection pursuit. Ann. Statist., 12, no. 3, p. 793-815 (1984). | MR 751274 | Zbl 0559.62002

[14] Donaldson (S.).— Kähler geometry on toric manifolds, and some other manifolds with large symmetry. Handbook of geometric analysis. Adv. Lect. Math. (ALM), 7, Int. Press, Somerville, MA, p. 29-75 (2008). | MR 2483362 | Zbl 1161.53066

[15] Eldan (R.), Klartag (B.).— Approximately gaussian marginals and the hyperplane conjecture. Proc. of a workshop on “Concentration, Functional Inequalities and Isoperimetry”, Contermporary Math., 545, Amer. Math. Soc., p. 55-68 (2011). | MR 2858465 | Zbl 1235.52012

[16] Escobar (J.).— Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities, and an eigenvalue estimate. Comm. Pure Appl. Math., 43, no. 7, p. 857-883 (1990). | MR 1072395 | Zbl 0713.53024

[17] Fleury (B.).— Inégalités de concentration pour les corps convexes. Thèse de Doctorat, Université Paris 6 (2009).

[18] Folland (G. B.).— Introduction to Partial Differential Equations. Mathematical Notes, Princeton University Press, Princeton, NJ (1976). | MR 599578 | Zbl 0841.35001

[19] Gilbarg (D.), Trudinger (N.).— Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin (2001). | MR 1814364 | Zbl 1042.35002

[20] Gromov (M.).— Convex sets and Kähler manifolds. Advances in differential geometry and topology, World Sci. Publ., Teaneck, NJ, p. 1-38 (1990). | MR 1095529 | Zbl 0770.53042

[21] Guillemin (V.).— Kähler structures on toric varieties. J. Diff. Geom., 40, p. 285-309 (1994). | MR 1293656 | Zbl 0813.53042

[22] Klartag (B.).— A central limit theorem for convex sets. Invent. Math. 168, no. 1, p. 91-131 (2007). | MR 2285748 | Zbl 1144.60021

[23] Klartag (B.).— A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Related Fields 145, no. 1-2, p. 1-33 (2009). | MR 2520120 | Zbl 1171.60322

[24] Klartag (B.).— High-dimensional distributions with convexity properties. Proc. of the Fifth Euro. Congress of Math., Amsterdam, July 2008. Eur. Math. Soc. publishing house, p. 401-417 (2010). | MR 2648334 | Zbl 1202.46011

[25] Kolesnikov (A.).— Hessian metrics and optimal transportation of log-concave measures. Preprint. Available under http://arxiv.org/abs/1201.2342

[26] Müller (C.).— Spherical harmonics. Lecture Notes in Math., 17, Springer-Verlag, Berlin-New York (1966). | MR 199449 | Zbl 0138.05101

[27] Petersen (P.).— Riemannian geometry. Second edition. Graduate Texts in Mathematics, 171. Springer, New York (2006). | MR 2243772 | Zbl 1220.53002

[28] Sudakov (V. N.).— Typical distributions of linear functionals in finite-dimensional spaces of high-dimension. (Russian) Dokl. Akad. Nauk. SSSR, 243, no. 6, (1978), 1402–1405. English translation in Soviet Math. Dokl., 19, p. 1578-1582 (1978). | MR 517198 | Zbl 0416.60005

[29] Tian (G.).— Canonical metrics in Kähler geometry. Notes taken by Meike Akveld. Lectures in Mathematics, ETH Zürich. Birkhäuser Verlag, Basel (2000). | MR 1787650 | Zbl 0978.53002

[30] Villani (C.).— Topics in optimal transportation. Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI (2003). | MR 1964483 | Zbl 1106.90001

[31] von Weizsäcker (H.).— Sudakov’s typical marginals, random linear functionals and a conditional central limit theorem. Probab. Theory and Related Fields, 107, no. 3, p. 313-324 (1997). | MR 1440135 | Zbl 0868.60009