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

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.

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.

DOI : 10.5802/afst.1366
Klartag, Bo’az 1

1 School of Mathematical Sciences, Tel-Aviv University, Tel Aviv 69978, Israel
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Klartag, Bo’az. Poincaré Inequalities and Moment Maps. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 22 (2013) no. 1, pp. 1-41. doi : 10.5802/afst.1366. http://www.numdam.org/articles/10.5802/afst.1366/

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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). | EuDML | Numdam | MR | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

[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 | Zbl

[25] Kolesnikov (A.).— Hessian metrics and optimal transportation of log-concave measures. Preprint. Available under | arXiv

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

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

[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 | Zbl

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

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

[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 | Zbl

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