Inégalités isopérimétriques en analyse et probabilités

Michel Ledoux

Séminaire Bourbaki (1992-1993)

  • Volume: 35, page 343-375
  • ISSN: 0303-1179

How to cite

top

Ledoux, Michel. "Inégalités isopérimétriques en analyse et probabilités." Séminaire Bourbaki 35 (1992-1993): 343-375. <http://eudml.org/doc/110174>.

@article{Ledoux1992-1993,
author = {Ledoux, Michel},
journal = {Séminaire Bourbaki},
keywords = {isoperimetric inequalities; concentration measures},
language = {fre},
pages = {343-375},
publisher = {Société Mathématique de France},
title = {Inégalités isopérimétriques en analyse et probabilités},
url = {http://eudml.org/doc/110174},
volume = {35},
year = {1992-1993},
}

TY - JOUR
AU - Ledoux, Michel
TI - Inégalités isopérimétriques en analyse et probabilités
JO - Séminaire Bourbaki
PY - 1992-1993
PB - Société Mathématique de France
VL - 35
SP - 343
EP - 375
LA - fre
KW - isoperimetric inequalities; concentration measures
UR - http://eudml.org/doc/110174
ER -

References

top
  1. [Az] K. Azuma. Weighted sums of certain dependent random variables. Tohoku Math. J.19, 357-367 (1967). Zbl0178.21103MR221571
  2. [Bae] A. Baernstein II. Integral means, univalent functions and circular symmetrization. Acta Math.133, 139-169 (1974). Zbl0315.30021MR417406
  3. [B-T] A. Baernstein II, B.A. Taylor. Spherical rearrangements, subharmonic functions and *-functions in n-space. Duke Math. J.43, 245-268 (1976). Zbl0331.31002MR402083
  4. [Ba] D. Bakry. L'hypercontractivité et son utilisation en théorie des semigroupes. École d'Été de Probabilités de St-Flour1992. Lecture Notes in Math.. Springer-Verlag. A paraître. Zbl0856.47026
  5. [B-É] D. Bakry, M. Émery. Diffusions hypercontractives. Séminaire de ProbabilitésXIX. Lecture Notes in Math.1123, 175-206 (1985). Springer-Verlag. Zbl0561.60080MR889476
  6. [Be] W. Beckner. Inequalities in Fourier analysis. Ann. Math.102,159-182 (1975). Zbl0338.42017MR385456
  7. [BA-L] G. BenArous, M. Ledoux. Schilder's large deviation principle without topology. Asymptotic problems in probability theory : Wiener functionals and asymptotics. Pitman Research Notes in Math. Series 284, 107-121 (1993). Longman. Zbl0801.60019MR1354164
  8. [Ben] Y. Benyamini. Two point symmetrization, the isoperimetric inequality on the sphere and some applications. Texas Functional Analysis Seminar1983-84. The University of Texas (1984). MR832231
  9. [B-B-G] P. Bérard, G. Besson, S. Gallot. Sur une inégalité isopérimétrique qui généralise celle de Paul Lévy-Gromov. Invent. Math.80, 295-308 (1985). Zbl0571.53027MR788412
  10. [B-G-M] M. Berger, P. Gauduchon, E. Mazet. Le spectre d'une variété riemannienne. Lecture Notes in Math.194 (1971). Springer-Verlag. Zbl0223.53034MR282313
  11. [Bo1] C. Borell. The Brunn-Minskowski inequality in Gauss space. Invent. Math.30, 207-216 (1975). Zbl0292.60004MR399402
  12. [Bo2] C. Borell. Tail probabilities in Gauss space. Vector Space Measures and Applications, Dublin1977. Lecture Notes in Math.644, 71-82 (1978). Springer-Verlag. Zbl0397.60015MR502400
  13. [Bo3] C. Borell. On polynomials chaos and integrability. Prob. Math. Statist.3, 191-203 (1984). Zbl0555.60008MR764146
  14. [Bo4] C. Borell. Geometric bounds on the Ornstein-Uhlenbeck process. Z. Wahrscheinlichkeitstheor. verw. Gebiete70, 1-13 (1985). Zbl0537.60084MR795785
  15. [B-Z] Y.D. Burago, V.A. Zalgaller. Geometric inequalities. Springer-Verlag (1988). Première édition (en russe) : Nauka (1980). Zbl0633.53002MR936419
  16. [Bu] P. Buser. A note on the isoperimetric constant. Ann. scient. Éc. Norm. Sup.15, 213-230 (1982). Zbl0501.53030MR683635
  17. [C-F] I. Chavel, E. Feldman. Modified isoperimetric constants, and large time heat diffusion in Riemannian manifold. Duke Math. J.64, 473-499 (1991). Zbl0753.58031MR1141283
  18. [C-L-Y] S. Cheng, P. Li, S.-T. Yau. On the upper estimate of the heat kernel on a complete Riemannian manifold. Amer. J. Math.156, 153-201 (1986). Zbl0611.58045MR630777
  19. [Ch] S. Chevet. Gaussian measures and large deviations. Probability in Banach spaces IV. Lecture Notes in Math.990, 30-46 (1983). Springer-Verlag. Zbl0527.60005MR707507
  20. [Co] T. Coulhon. Sobolev inequalities on graphs and on manifolds. Harmonic Analysis and Discrete Potential Theory, 207-214 (1991). Plenum Press. MR1222459
  21. [C-L] T. Coulhon, M. Ledoux. Isopérimétrie, décroissance du noyau de la chaleur et transformées de Riesz : un contre-exemple (1992). À paraître in Ark. för Math.. Zbl0826.53035
  22. [C-SC] T. Coulhon, L. Saloff-Coste. Isopérimétrie pour les groupes et les variétés (1992). À paraître in Rev. Ibero Americana. Zbl0782.53066MR1232845
  23. [Da] E.B. Davies. Heat kernels and spectral theory. Cambridge Univ. Press (1989). Zbl0699.35006MR990239
  24. [Dv] A. Dvoretzky. Some results on convex bodies and Banach spaces. Proc. Symp. on Linear Spaces, Jerusalem, 123-160 (1961). Zbl0119.31803MR139079
  25. [D-F] P. Diaconis, D. Freedman. A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincaré23, 397-423 (1987). Zbl0619.60039MR898502
  26. [Eh1] A. Ehrhard. Symétrisation dans l'espace de Gauss. Math. Scand.53, 281-301 (1983). Zbl0542.60003MR745081
  27. [Eh2] A. Ehrhard. Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes. Ann. scient. Éc. Norm. Sup.17, 317-332 (1984). Zbl0546.49020MR760680
  28. [Eh3] A. Ehrhard. Eléments extrémaux pour les inégalités de Brunn-Minskowski gaussiennes. Ann. Inst. H. Poincaré22, 149-168 (1986). Zbl0595.60020MR850753
  29. [Eh4] A. Ehrhard. Sur l'inégalité de Sobolev logarithmique de Gross. Séminaire de ProbabilitésXVIII. Lecture Notes in Math.1059, 194-196 (1984). Springer-Verlag. Zbl0542.60040MR770961
  30. [Fa1] S. Fang. Isoperimetric inequalities on the Wiener space. Bull. Sc. Math.112, 345-355 (1988). Zbl0689.60026MR975368
  31. [Fa2] S. Fang. Grandes déviations pour le processus d'Ornstein-Uhlenbeck. Prépublication (1991). 
  32. [Fe] H. Federer. Geometric measure theory. Springer-Verlag (1969). Zbl0874.49001
  33. [Fer1] X. Fernique. Intégrabilité des vecteurs gaussiens. C. R. Acad. Sci.Paris270, 1698-1699 (1970). Zbl0206.19002MR266263
  34. [Fer2] X. Fernique. Fonctions aléatoires gaussiennes, les résultats récents de M. Talagrand. Séminaire Bourbaki, exp. 660. Astérisque145-146, 177-186 (1987). Zbl0623.60052MR880032
  35. [F-L-M] T. Figiel, J. Lindenstrauss, V.D. Milman. The dimensions of almost spherical sections of convex bodies. Acta Math.139, 52-94 (1977). Zbl0375.52002MR445274
  36. [G-H-L] S. Gallot, D. Hulin, J. Lafontaine. Riemannian Geometry. Deuxième édition, Springer-Verlag (1990). Zbl0636.53001MR1083149
  37. [Gr1] M. Gromov. Paul Lévy's isoperimetric inequality. Prépublication I.H.E.S. (1980). 
  38. [Gr2] M. Gromov. Groups of polynomial growth and expanding maps. Publ. Math. I.H.E.S.53, 53-78 (1981). Zbl0474.20018MR623534
  39. [G-M] M. Gromov, V.D. Milman. A topological application of the isoperimetric inequality. Amer. J. Math.105, 843-854 (1983). Zbl0522.53039MR708367
  40. [Gro] L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math.97, 1061-1083 (1975). Zbl0318.46049MR420249
  41. [Gu] Y. Guivarc'h. Croissance polynômiale et périodes des fonctions harmoniques. Bull. Soc. Math. France101, 333-379 (1973). Zbl0294.43003MR369608
  42. [Ha] L.H. Harper. Optimal numbering and isoperimetric problems on graphs. J. Comb. Th.1, 385-393 (1966). Zbl0158.20802MR200192
  43. [Ka] J.-P. Kahane. Some random series of functions. Heath Math. Monographs (1968). Deuxième édition : Cambridge Univ. Press (1985). Zbl0192.53801MR833073
  44. [Ke] H. Kesten. On the speed of convergence in first-passage percolation (1991). À paraître in Ann. Appl. Probability. Zbl0783.60103MR1221154
  45. [K-L] J. Kuelbs, W. Li. Metric entropy and the small ball problem for Gaussian measures (1992). À paraître in J. Funct. Anal.. Zbl0799.46053MR1184915
  46. [L-S] H.J. Landau, L.A. Shepp. On the supremum of a Gaussian process. Sankhyà A32, 369-378 (1970). Zbl0218.60039MR286167
  47. [Led1] M. Ledoux. A note on large deviations for Wiener chaos. Séminaire de ProbabilitésXXIV. Lecture Notes in Math.1426, 1-14, Springer-Verlag (1990). Zbl0701.60020MR1071528
  48. [Led2] M. Ledoux. A heat semigroup approach to concentration on the sphere and on a compact Riemannian manifold. Geometric and Funct. Anal.2, 221-224 (1992). Zbl0752.53039MR1159831
  49. [Led3] M. Ledoux. Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space (1992). À paraître in Bull. Sc. math.. Zbl0841.49024MR1309086
  50. [Led4] M. Ledoux. A simple analytic proof of an inequality by P. Buser (1992). À paraître in Proc. Amer. Math. Soc.. Zbl0812.58093MR1186991
  51. [L-T1] M. Ledoux, M. Talagrand. Characterization of the law of the iterated logarithm in Banach spaces. Ann. Probability16, 1242-1264 (1988). Zbl0662.60008MR942766
  52. [L-T2] M. Ledoux, M. Talagrand. Probability in Banach spaces (Isoperimetry and processes). Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag (1991). Zbl0748.60004MR1102015
  53. [Lé] P. Lévy. Problèmes concrets d'analyse fonctionnelle. Gauthier-Villars (1951). Zbl0043.32302MR41346
  54. [L-Y] P. Li, S.-T. Yau. On the parabolic kernel of the Schrödinger operator. Acta Math.156, 153-201 (1986). Zbl0611.58045MR834612
  55. [MD] C.J.H. McDiarmid. On the method of bounded differences. Twelfth British Combinatorial Conference. Surveys in Combinatorics, 148-188 (1989). Cambrige Univ. Press. Zbl0712.05012MR1036755
  56. [MK] H.P. McKean. Geometry of differential space. Ann. Probability1, 197-206 (1973). Zbl0263.60035MR353471
  57. [Ma1] B. Maurey. Constructions de suites symétriques. C. R. Acad. Sci. Paris288, 679-681 (1979). Zbl0398.46019MR533901
  58. [Ma2] B. Maurey. Sous-espaces lp des espaces de Banach. Séminaire Bourbaki, exp. 608. Astérisque105-106, 199-216 (1983). Zbl0553.46020MR728989
  59. [Ma3] B. Maurey. Some deviations inequalities. Geometric and Funct. Anal.1, 188- 197 (1991). Zbl0756.60018MR1097258
  60. [Mi1] V.D. Milman. New proof of the theorem of Dvoretzky on sections of convex bodies. Funct. Anal. Appl.5, 28-37 (1971). Zbl0239.46018MR293374
  61. [Mi2] V.D. Milman. The heritage of P. Lévy in geometrical functional analysis. Colloque Paul Lévy sur les processus stochastiques. Astérisque157-158, 273- 302 (1988). Zbl0681.46021MR976223
  62. [Mi3] V.D. Milman. Dvoretzky's theorem - Thirty years later (Survey). Geometric and Funct. Anal.2, 455-479 (1992). Zbl0787.46016MR1191569
  63. [M-S] V.D. Milman, G. Schechtman. Asymptotic theory of finite dimensional normed spaces. Lecture Notes in Math.1200 (1986). Springer-Verlag. Zbl0606.46013MR856576
  64. [Mo] J. Moser. On Harnack's theorem for elliptic differential equations. Comm. Pure Appl. Math.14, 557-591 (1961). Zbl0111.09302MR159138
  65. [N-S-W] A. Nagel, E. Stein, M. Wainger. Balls and metrics defined by vector fields. Acta Math.155, 103-147 (1985). Zbl0578.32044MR793239
  66. [Na] J. Nash. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math.80, 931-954 (1958). Zbl0096.06902MR100158
  67. [Ne] E. Nelson. The free Markov field. J. Funct. Anal.12, 211-227 (1973). Zbl0273.60079MR343816
  68. [Os] R. Osserman. The isoperimetric inequality. Bull. Amer. Math. Soc.84, 1182- 1238 (1978). Zbl0411.52006MR500557
  69. [Pi1] G. Pisier. Probabilistic methods in the geometry of Banach spaces. Probability and Analysis, Varenna (Italy)1985. Lecture Notes in Math.1206, 167-241 (1986). Springer-Verlag. Zbl0606.60008MR864714
  70. [Pi2] G. Pisier. Riesz transforms : a simpler analytic proof of P. A. Meyer inequality. Séminaire de ProbabilitésXXII. Lecture Notes in Math.1321, 485-501, Springer-Verlag (1988). Zbl0645.60061MR960544
  71. [Pi3] G. Pisier. The volume of convex bodies and Banach space geometry. Cambridge Univ. Press (1989). Zbl0698.46008MR1036275
  72. [SC1] L. Saloff-Coste. Analyse sur les groupes de Lie à croissance polynômiale. Ark. för Math.28, 315-331 (1990). Zbl0715.43009MR1084020
  73. [SC2] L. Saloff-Coste. A note on Poincaré, Sobolev and Harnack inequalities. Duke Math. J.65, 27-38 (1992). Zbl0769.58054MR1150597
  74. [Sc] E. Schmidt. Die Brunn-Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und nichteuklidischen Geometrie. Math. Nach.1, 81-157 (1948). Zbl0030.07602MR28600
  75. [S-T] V.N. Sudakov, B.S. Tsirel'son. Extremal properties of half-spaces for spherically invariant measures. J. Soviet. Math.9, 9-18 (1978) ; translated from Zap. Nauch. Sem. L.O.M.I.41, 14-24 (1974). Zbl0395.28007MR365680
  76. [Ta1] M. Talagrand. Sur l'intégrabilité des vecteurs gaussiens. Z. Wahrscheinlichkeitstheor. verw. Gebiete68, 1-8 (1984). Zbl0529.60034MR767440
  77. [Ta2] M. Talagrand. Regularity of Gaussian processes. Acta Math.159, 99-149 (1987). Zbl0712.60044MR906527
  78. [Ta3] M. Talagrand. An isoperimetric theorem on the cube and the Khintchine-Kahane inequalities. Proc. Amer. Math. Soc.104, 905-909 (1988). Zbl0691.60015MR964871
  79. [Ta4] M. Talagrand. Small tails for the supremum of a Gaussian process. Ann. Inst. H. Poincaré24, 307-315 (1988). Zbl0641.60044MR953122
  80. [Ta5] M. Talagrand. Isoperimetry and integrability of the sum of independent Banach space valued random variables. Ann. Probability17, 1546-1570 (1989). Zbl0692.60016MR1048946
  81. [Ta6] M. Talagrand. Supremum of some canonical processes (1990). À paraître in J. Amer. Math. Soc.. 
  82. [Ta7] M. Talagrand. A new isoperimetric inequality and the concentration of measure phenomenon. Geometric Aspects of Functional Analysis, Israel Seminar1989-90. Lecture Notes in Math.1469, 94-124 (1991). Springer-Verlag. Zbl0818.46047MR1122615
  83. [Ta8] M. Talagrand. Simple proof of the majorizing measure theorem. Geometric and Funct. Anal.2, 118-125 (1992). Zbl0760.60043MR1143666
  84. [Ta9] M. Talagrand. A new isoperimetric inequality for product measure and the tails of sums of independent random variables. Geometric and Funct. Anal.1, 211-223 (1991). Zbl0760.60005MR1097260
  85. [Ta10] M. Talagrand. On the rate of clustering in Strassen's law of the iterated logarithm. Probability in Banach spaces 8. Progress in Probability30, 339- 351 (1992). Birkhäuser. MR1227629
  86. [Ta11] M. Talagrand. Sharper bounds for Gaussian and empirical processes (1992). À paraître in Ann. Probability. Zbl0798.60051
  87. [Ta12] M. Talagrand. New Gaussian estimates for enlarged balls (1992). À paraître in Geometric and Funct. Anal.. Zbl0815.46021MR1233864
  88. [Ta13] M. Talagrand. Isoperimetry, logarithmic Sobolev inequalities on the discrete cube, and Margulis' graph connectivity theorem (1992). À paraître. Zbl0806.46035MR1215783
  89. [Ta14] M. Talagrand. Regularity of infinitely divisible processes. Ann. Probability21, 362-432 (1993). Zbl0776.60053MR1207231
  90. [Ta15] M. Talagrand. Concentration of measure and isoperimetric inequalities in product spaces. En préparation (1993). Zbl0864.60013
  91. [Va1] N. Varopoulos. Une généralisation du théorème de Hardy-Littlewood-Sobolev pour les espaces de Dirichlet. C. R. Acad. Sci. Paris299, 651-654 (1984). Zbl0566.31006MR770455
  92. [Va2] N. Varopoulos. Hardy-Littlewood theory for semigroups. J. Funct. Anal.63, 240-260 (1985). Zbl0608.47047MR803094
  93. [Va3] N. Varopoulos. Isoperimetric inequalities and Markov chains. J. Funct. Anal.63, 215-239 (1985). Zbl0573.60059MR803093
  94. [Va4] N. Varopoulos. Analysis on Lie groups. J. Funct. Anal.76, 346-410 (1988). Zbl0634.22008MR924464
  95. [Va5] N. Varopoulos. Small time Gaussian estimates of heat diffusion kernels. Part I : The semigroup technique. Bull. Sc. math.113, 253-277 (1989). Zbl0703.58052MR1016211
  96. [Va6] N. Varopoulos. Analysis and geometry on groups. Proceedings of the International Congress of Mathematicians, Kyoto (1990), vol. II, 951-957 (1991). Springer-Verlag. Zbl0744.43006MR1159280
  97. [V-SC-C] N. Varopoulos, L. Saloff-Coste, T. Coulhon. Analysis and geometry on groups. Cambridge Univ. Press (1992). Zbl0813.22003MR1218884
  98. [W-W] D.L. Wang, P. Wang. Extremal configurations on a discrete torus and a generalization of the generalized Macaulay theorem. Siam J. Appl. Math.33, 55-59 (1977). Zbl0362.05048MR438237

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.