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 Electronic Journal of Probability > Vol. 12 (2007) > Paper 44 open journal systems 


Isoperimetry between exponential and Gaussian

Franck Barthe, Institut de Mathematiques
Patrick Cattiaux, Ecole Polytechnique et MODALX
Cyril Roberto, Université de Marne La Vallée


Abstract
We study the isoperimetric problem for product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate half-spaces are approximate solutions of the isoperimetric problem


Full text: PDF

Pages: 1212-1237

Published on: September 12, 2007


Bibliography
  1. Aida, Shigeki. Uniform positivity improving property, Sobolev inequalities, and spectral gaps. J. Funct. Anal. 158 (1998), no. 1, 152--185. MR1641566 (2000d:60125)
  2. C.Ane, S.Blachere, D.Chafai, P.Fougeres, I.Gentil, F.Malrieu, C.Roberto, and G.Scheffer. Sur les inegalites de Sobolev logarithmiques., volume~10 of Panoramas et Syntheses. S.M.F., Paris, 2000.
  3. Bakry, D.; Émery, Michel. Diffusions hypercontractives. (French) [Hypercontractive diffusions] Séminaire de probabilités, XIX, 1983/84, 177--206, Lecture Notes in Math., 1123, Springer, Berlin, 1985. MR0889476 (88j:60131)
  4. Bakry, D.; Ledoux, M. Lévy-Gromov's isoperimetric inequality for an infinite-dimensional diffusion generator. Invent. Math. 123 (1996), no. 2, 259--281. MR1374200 (97c:58162)
  5. Barthe, Franck. Levels of concentration between exponential and Gaussian. Ann. Fac. Sci. Toulouse Math. (6) 10 (2001), no. 3, 393--404. MR1923685 (2003f:60038)
  6. Barthe, F. Log-concave and spherical models in isoperimetry. Geom. Funct. Anal. 12 (2002), no. 1, 32--55. MR1904555 (2003d:28017)
  7. Barthe, F. Infinite dimensional isoperimetric inequalities in product spaces with the supremum distance. J. Theoret. Probab. 17 (2004), no. 2, 293--308. MR2053705 (2005f:28032)
  8. F.Barthe, P.Cattiaux, and C.Roberto. Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and application to isoperimetry. Revista Mat. Iberoamericana, To appear.
  9. Barthe, F.; Roberto, C. Sobolev inequalities for probability measures on the real line. Dedicated to Professor Aleksander Pelczynski on the occasion of his 70th birthday (Polish). Studia Math. 159 (2003), no. 3, 481--497. MR2052235 (2006c:60019)
  10. Bertini, Lorenzo; Zegarlinski, Bogusl aw. Coercive inequalities for Gibbs measures. J. Funct. Anal. 162 (1999), no. 2, 257--286. MR1682059 (2000c:60158)
  11. Bobkov, S. Extremal properties of half-spaces for log-concave distributions. Ann. Probab. 24 (1996), no. 1, 35--48. MR1387625 (97e:60027)
  12. Bobkov, S. G. Isoperimetric problem for uniform enlargement. Studia Math. 123 (1997), no. 1, 81--95. MR1438305 (98d:60006)
  13. Bobkov, S. G.; Götze, F. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1999), no. 1, 1--28. MR1682772 (2000b:46059)
  14. Bobkov, S. G.; Udre, K. Characterization of Gaussian measures in terms of the isoperimetric property of half-spaces. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 228 (1996), Veroyatn. i Stat. 1, 31--38, 356; translation in J. Math. Sci. (New York) 93 (1999), no. 3, 270--275 MR1449845 (98e:60056)
  15. Bobkov, S. G.; Houdré, C. Isoperimetric constants for product probability measures. Ann. Probab. 25 (1997), no. 1, 184--205. MR1428505 (98g:60032)
  16. Bobkov, Sergey G.; Houdré, Christian. Weak dimension-free concentration of measure. Bernoulli 6 (2000), no. 4, 621--632. MR1777687 (2001i:28005)
  17. Bobkov, Serguei G.; Houdré, Christian. Some connections between isoperimetric and Sobolev-type inequalities. Mem. Amer. Math. Soc. 129 (1997), no. 616, viii+111 pp. MR1396954 (98b:46038)
  18. Bobkov, S. G.; Zegarlinski, B. Entropy bounds and isoperimetry. Mem. Amer. Math. Soc. 176 (2005), no. 829, x+69 pp. MR2146071 (2006c:46027)
  19. Borell, Christer. The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30 (1975), no. 2, 207--216. MR0399402 (53 #3246)
  20. Borell, Christer. Intrinsic bounds for some real-valued stationary random functions. Probability in Banach spaces, V (Medford, Mass., 1984), 72--95, Lecture Notes in Math., 1153, Springer, Berlin, 1985. MR0821977 (87h:60079)
  21. Cattiaux, Patrick. Hypercontractivity for perturbed diffusion semigroups. Ann. Fac. Sci. Toulouse Math. (6) 14 (2005), no. 4, 609--628. MR2188585 (2006i:60112)
  22. Davies, E. B. Heat kernels and spectral theory. Cambridge Tracts in Mathematics, 92. Cambridge University Press, Cambridge, 1989. x+197 pp. ISBN: 0-521-36136-2 MR0990239 (90e:35123)
  23. Federer, Herbert. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969 xiv+676 pp. MR0257325 (41 #1976)
  24. Fougères, Pierre. Hypercontractivité et isopérimétrie gaussienne. Applications aux systèmes de spins. (French) [Hypercontractivity and Gaussian isoperimetry. Applications to spin systems] Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 5, 647--689. MR1792659 (2001m:82004)
  25. Gong, Fu-Zhou; Wang, Feng-Yu. Functional inequalities for uniformly integrable semigroups and application to essential spectrums. Forum Math. 14 (2002), no. 2, 293--313. MR1880915 (2003a:47097)
  26. Helffer, Bernard. Semiclassical analysis, Witten Laplacians, and statistical mechanics. Series in Partial Differential Equations and Applications, 1. World Scientific Publishing Co., Inc., River Edge, NJ, 2002. x+179 pp. ISBN: 981-238-098-1 MR1936110 (2003j:58038)
  27. Holley, Richard; Stroock, Daniel. Logarithmic Sobolev inequalities and stochastic Ising models. J. Statist. Phys. 46 (1987), no. 5-6, 1159--1194. MR0893137 (89e:82013)
  28. Kwapien, S.; Pycia, M.; Schachermayer, W. A proof of conjecture of Bobkov and Houdre. Electron. Comm. Probab. 1 (1996), no. 2, 7--10 (electronic). MR1386289 (97c:60032)
  29. Latal a, R.; Oleszkiewicz, K. Between Sobolev and Poincaré. Geometric aspects of functional analysis, 147--168, Lecture Notes in Math., 1745, Springer, Berlin, 2000. MR1796718 (2002b:60025)
  30. Ledoux, M. A simple analytic proof of an inequality by P. Buser. Proc. Amer. Math. Soc. 121 (1994), no. 3, 951--959. MR1186991 (94i:53041)
  31. Ledoux, Michel. Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités, XXXIII, 120--216, Lecture Notes in Math., 1709, Springer, Berlin, 1999. MR1767995 (2002j:60002)
  32. Ledoux, Michel. The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, RI, 2001. x+181 pp. ISBN: 0-8218-2864-9 MR1849347 (2003k:28019)
  33. V.G. Maz'ja. Sobolev spaces. Springer Series in Soviet Mathematics. Springer, Berlin, 1985.
  34. Oleszkiewicz, Krzysztof. On certain characterization of normal distribution. Statist. Probab. Lett. 33 (1997), no. 3, 277--280. MR1456703 (98i:60015)
  35. Roberto, C.; Zegarli'nski, B. Orlicz-Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semi-groups. J. Funct. Anal. 243 (2007), no. 1, 28--66. MR2289793
  36. Röckner, Michael; Wang, Feng-Yu. Weak Poincaré inequalities and $Lsp 2$-convergence rates of Markov semigroups. J. Funct. Anal. 185 (2001), no. 2, 564--603. MR1856277 (2002j:47075)
  37. Ros, Antonio. The isoperimetric problem. Global theory of minimal surfaces, 175--209, Clay Math. Proc., 2, Amer. Math. Soc., Providence, RI, 2005. MR2167260 (2006e:53023)
  38. Rothaus, O. S. Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities. J. Funct. Anal. 64 (1985), no. 2, 296--313. MR0812396 (87f:58181)
  39. Sudakov, V. N.; Cirelcprime son, B. S. Extremal properties of half-spaces for spherically invariant measures. (Russian) Problems in the theory of probability distributions, II. Zap. Nauv cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 (1974), 14--24, 165. MR0365680 (51 #1932)
  40. Talagrand, Michel. A new isoperimetric inequality and the concentration of measure phenomenon. Geometric aspects of functional analysis (1989--90), 94--124, Lecture Notes in Math., 1469, Springer, Berlin, 1991. MR1122615 (93d:60095)
  41. Talagrand, Michel. Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. No. 81 (1995), 73--205. MR1361756 (97h:60016)
  42. Wang, Feng-Yu. Functional inequalities for empty essential spectrum. J. Funct. Anal. 170 (2000), no. 1, 219--245. MR1736202 (2001a:58043)
  43. Wang, Feng-Yu. A generalization of Poincaré and log-Sobolev inequalities. Potential Anal. 22 (2005), no. 1, 1--15. MR2127729 (2006a:60031)
















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Electronic Journal of Probability. ISSN: 1083-6489