G. Bennett, Probability Inequalities for the Sum of Independent Random Variables, Journal of the American Statistical Association, vol.18, issue.297, pp.33-45, 1962.
DOI : 10.6028/NBS.RPT.1744

S. Bobkov, M. Fradelizi, and M. Madiman, When can one invert Hölder's inequality? (and why one may want to), 2015.

S. Bobkov and M. Madiman, Concentration of the information in data with log-concave distributions, The Annals of Probability, vol.39, issue.4, pp.1528-1543, 2011.
DOI : 10.1214/10-AOP592

S. Bobkov and M. Madiman, Dimensional behaviour of entropy and information, Comptes Rendus Mathematique, vol.349, issue.3-4, pp.201-204, 2011.
DOI : 10.1016/j.crma.2011.01.008

URL : http://arxiv.org/pdf/1101.3352

S. Bobkov and M. Madiman, The Entropy Per Coordinate of a Random Vector is Highly Constrained Under Convexity Conditions, IEEE Transactions on Information Theory, vol.57, issue.8, pp.4940-4954, 2011.
DOI : 10.1109/TIT.2011.2158475

S. Bobkov and M. Madiman, Reverse Brunn???Minkowski and reverse entropy power inequalities for convex measures, Journal of Functional Analysis, vol.262, issue.7, pp.3309-3339, 2012.
DOI : 10.1016/j.jfa.2012.01.011

URL : http://doi.org/10.1016/j.jfa.2012.01.011

F. Bolley, I. Gentil, and A. Guillin, Dimensional improvements of the logarithmic sobolev, talagrand and brascamp-lieb inequalities, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01171361

C. Borell, Complements of Lyapunov's inequality, Mathematische Annalen, vol.21, issue.4, pp.323-331, 1973.
DOI : 10.1007/BF01362702

S. Boucheron, G. Lugosi, and P. Massart, Concentration inequalities A nonasymptotic theory of independence
URL : https://hal.archives-ouvertes.fr/hal-00794821

S. Boyd and L. Vandenberghe, Convex optimization, 2004.

D. Cordero-erausquin, M. Fradelizi, G. Paouris, and P. Pivovarov, Volume of the polar of random sets and shadow systems, Mathematische Annalen, vol.2, issue.3, 2014.
DOI : 10.1007/BF02759738

URL : https://hal.archives-ouvertes.fr/hal-01122831

R. Eldan, Thin Shell Implies Spectral Gap Up to Polylog via a Stochastic Localization Scheme, Geometric and Functional Analysis, vol.272, issue.5, pp.532-569, 2013.
DOI : 10.1090/gsm/058

R. Eldan and B. Klartag, Approximately Gaussian marginals and the hyperplane conjecture, Concentration, Functional Inequalities and Isoperimetry, pp.55-68, 2011.
DOI : 10.1090/conm/545/10764

URL : http://arxiv.org/abs/1001.0875

R. Eldan and J. Lehec, Bounding the Norm of a Log-Concave Vector Via Thin-Shell Estimates, Lecture Notes in Mathematics, vol.2116, pp.107-122, 2014.
DOI : 10.1007/978-3-319-09477-9_9

URL : https://hal.archives-ouvertes.fr/hal-01100946

M. Fradelizi, Sections of convex bodies through their centroid, Archiv der Mathematik, vol.69, issue.6, pp.515-522, 1997.
DOI : 10.1007/s000130050154

M. Fradelizi, J. Li, and M. Madiman, Concentration of information content and other functionals under convex measures, 2015.

M. Fradelizi and M. Meyer, Increasing functions and inverse Santal?? inequality for unconditional functions, Positivity, vol.12, issue.3, pp.407-420, 2008.
DOI : 10.1007/s11117-007-2145-z

O. Guédon and E. Milman, Interpolating Thin-Shell and Sharp Large-Deviation Estimates for Lsotropic Log-Concave Measures, Geometric and Functional Analysis, vol.16, issue.5, pp.1043-1068, 2011.
DOI : 10.1007/s00039-006-0584-5

G. Hargé, Reinforcement of an inequality due to Brascamp and Lieb, Journal of Functional Analysis, vol.254, issue.2, pp.267-300, 2008.
DOI : 10.1016/j.jfa.2007.07.019

J. Hiriart-urruty and C. Lemaréchal, Convex analysis and minimization algorithms. I, volume 305 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences, 1993.
DOI : 10.1007/978-3-662-02796-7

B. Klartag, A central limit theorem for convex sets, Inventiones mathematicae, vol.107, issue.3, pp.91-131, 2007.
DOI : 10.1016/j.crma.2005.11.018

URL : http://arxiv.org/abs/math/0605014

B. Klartag, Eigenvalue distribution of optimal transportation, Presentation given at the Workshop on Information Theory and Concentration Phenomena held at the Institute for Mathematics and its Applications (IMA), 2015.
DOI : 10.1073/pnas.35.7.408

URL : http://arxiv.org/pdf/1402.2636

B. Klartag and A. Kolesnikov, Eigenvalue distribution of optimal transportation, Analysis & PDE, vol.7, issue.1, pp.33-55, 2015.
DOI : 10.1073/pnas.35.7.408

B. Klartag and V. D. Milman, Geometry of Log-concave Functions and Measures, Geometriae Dedicata, vol.302, issue.3, pp.169-182, 2005.
DOI : 10.1007/s10711-004-2462-3

M. Madiman, L. Wang, and S. Bobkov, Some applications of the nonasymptotic equipartition property of log-concave distributions, 2015.

V. H. Nguyen, Inégalités fonctionnelles et convexité, 2013.

V. H. Nguyen, Dimensional variance inequalities of Brascamp???Lieb type and a local approach to dimensional Pr??kopa??s theorem, Journal of Functional Analysis, vol.266, issue.2, pp.931-955, 2014.
DOI : 10.1016/j.jfa.2013.11.003

A. Prékopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged), vol.34, pp.335-343, 1973.

R. T. Rockafellar, Convex analysis. Princeton Mathematical Series, 1970.

A. Van-der-vaart and J. A. Wellner, A local maximal inequality under uniform entropy, Electronic Journal of Statistics, vol.5, issue.0, pp.192-203, 2011.
DOI : 10.1214/11-EJS605

L. Wang, Heat capacity bound, energy fluctuations and convexity, 2014.

L. Wang and M. Madiman, Beyond the Entropy Power Inequality, via Rearrangements, IEEE Transactions on Information Theory, vol.60, issue.9, pp.5116-5137, 2014.
DOI : 10.1109/TIT.2014.2338852

URL : http://arxiv.org/pdf/1307.6018

J. A. Wellner, Limit theorems for the ratio of the empirical distribution function to the true distribution function, Zeitschrift f???r Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol.5, issue.No. 2, pp.73-88, 1978.
DOI : 10.1007/BF00635964