A. D. Alexandroff, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, (Russian) Leningrad State Univ, Annals [Uchenye Zapiski] Math. Ser, vol.6, pp.3-35, 1939.

B. Andrews, Gauss curvature flow: the fate of the rolling stones, Inventiones Mathematicae, vol.138, issue.1, pp.151-161, 1999.
DOI : 10.1007/s002220050344

S. Artstein-avidan, B. Klartag, and V. Milman, The Santal?? point of a function, and a functional form of the Santal?? inequality, Mathematika, vol.10, issue.1-2, pp.33-48, 2004.
DOI : 10.1007/BF02018814

S. Artstein-avidan, B. Klartag, C. Schütt, and E. Werner, Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality, Journal of Functional Analysis, vol.262, issue.9, pp.4181-4204, 2012.
DOI : 10.1016/j.jfa.2012.02.014

URL : http://arxiv.org/abs/1110.5551

S. Artstein-avidan and V. , Milman The concept of duality in convex analysis, and the characterization of the Legendre transform, Ann. of Math, vol.2, pp.169-661, 2009.

D. Bakry and M. Ledoux, A logarithmic Sobolev form of the Li-Yau parabolic inequality, Revista Matem??tica Iberoamericana, vol.22, pp.683-702, 2006.
DOI : 10.4171/RMI/470

URL : https://hal.archives-ouvertes.fr/hal-00353946

K. Ball, Isometric problems in p and sections of convex sets, 1986.

C. Borell, Convex set functions ind-space, Periodica Mathematica Hungarica, vol.1, issue.2, pp.111-136, 1975.
DOI : 10.1214/aoms/1177728435

H. Busemann, W. Feller, and . Kruemmungseigenschaften-konvexer-flächen, Kr??mmungseigenschaften Konvexer Fl??chen, Acta Mathematica, vol.66, issue.0, pp.1-47, 1935.
DOI : 10.1007/BF02546515

U. Caglar and E. M. Werner, Divergence for s-concave and log concave functions, Advances in Mathematics, vol.257, pp.219-247, 2014.
DOI : 10.1016/j.aim.2014.02.013

U. Caglar and E. M. Werner, Mixed f -divergence and inequalities for logconcave functions, arxiv.org/abs/1401, p.7065
DOI : 10.1112/plms/pdu055

URL : http://arxiv.org/abs/1401.7065

G. Q. Chen, M. Torres, and W. P. Ziemer, Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, Communications on Pure and Applied Mathematics, vol.120, issue.3, pp.242-304, 2009.
DOI : 10.1002/cpa.20262

A. Cianchi, E. Lutwak, D. Yang, and G. Zhang, Affine Moser???Trudinger and Morrey???Sobolev inequalities, Calculus of Variations and Partial Differential Equations, vol.53, issue.3, pp.419-436, 2009.
DOI : 10.1007/s00526-009-0235-4

E. and D. Giorgi, Nuovi teoremi relativi alle misure (r-1)-dimensionali in uno spazio ad r dimensioni, Italian) Ricerche Mat, pp.95-113, 1955.

P. Federbush, Partially Alternate Derivation of a Result of Nelson, Journal of Mathematical Physics, vol.10, issue.1, pp.50-52, 1969.
DOI : 10.1063/1.1664760

H. Federer, A note on the Gauss-Green theorem, Proc. Amer, pp.447-451, 1958.
DOI : 10.1090/S0002-9939-1958-0095245-2

H. Federer, Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, 1969.

M. Fradelizi and M. Meyer, Some functional forms of Blaschke???Santal?? inequality, Mathematische Zeitschrift, vol.109, issue.2, pp.379-395, 2007.
DOI : 10.1007/s00209-006-0078-z

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

R. J. Gardner and G. Zhang, Affine inequalities and radial mean bodies, American Journal of Mathematics, vol.120, issue.3, pp.505-528, 1998.
DOI : 10.1353/ajm.1998.0021

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=

L. Gross, Logarithmic Sobolev Inequalities, American Journal of Mathematics, vol.97, issue.4, pp.1061-1083, 1975.
DOI : 10.2307/2373688

C. Haberl and F. Schuster, General $L_p$ affine isoperimetric inequalities, Journal of Differential Geometry, vol.83, issue.1, pp.1-26, 2009.
DOI : 10.4310/jdg/1253804349

URL : http://arxiv.org/abs/0809.1983

C. Haberl, F. E. Schuster, and J. Xiao, An asymmetric affine P??lya???Szeg?? principle, Mathematische Annalen, vol.53, issue.3, pp.517-542, 2012.
DOI : 10.1007/s00208-011-0640-9

URL : http://arxiv.org/abs/0908.1557

D. Hug, Curvature Relations and Affine Surface Area for a General Convex Body and its Polar, Results in Mathematics, vol.66, issue.308, pp.233-248, 1996.
DOI : 10.1007/BF03322221

B. Klartag and V. 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

J. Lehec, A direct proof of the functional Santal?? inequality, Comptes Rendus Mathematique, vol.347, issue.1-2, pp.55-58, 2009.
DOI : 10.1016/j.crma.2008.11.015

J. Lehec, Partitions and functional Santal?? inequalities, Archiv der Mathematik, vol.92, issue.1, pp.89-94, 2009.
DOI : 10.1007/s00013-008-3014-0

URL : http://arxiv.org/abs/1011.2119

M. Ludwig and M. Reitzner, A Characterization of Affine Surface Area, Advances in Mathematics, vol.147, issue.1, pp.138-172, 1999.
DOI : 10.1006/aima.1999.1832

M. Ludwig and M. Reitzner, ) invariant valuations, Annals of Mathematics, vol.172, issue.2, pp.1223-1271, 2010.
DOI : 10.4007/annals.2010.172.1223

M. Ludwig, J. Xiao, and G. Zhang, Sharp convex Lorentz???Sobolev inequalities, Mathematische Annalen, vol.53, issue.5, pp.169-197, 2011.
DOI : 10.1007/s00208-010-0555-x

E. Lutwak, The Brunn???Minkowski???Firey Theory II, Advances in Mathematics, vol.118, issue.2, pp.244-294, 1996.
DOI : 10.1006/aima.1996.0022

E. Lutwak, D. Yang, and G. Zhang, Lp Affine Isoperimetric Inequalities, Journal of Differential Geometry, vol.56, issue.1, pp.111-132, 2000.
DOI : 10.4310/jdg/1090347527

E. Lutwak, D. Yang, and G. Zhang, Sharp Affine LP Sobolev Inequalities, Journal of Differential Geometry, vol.62, issue.1, pp.17-38, 2002.
DOI : 10.4310/jdg/1090425527

E. Lutwak, D. Yang, and G. Zhang, Optimal Sobolev norms and the Lp Minkowski problem, International Mathematics Research Notices, pp.1-21, 2006.
DOI : 10.1155/IMRN/2006/62987

R. J. Mccann, A Convexity Principle for Interacting Gases, Advances in Mathematics, vol.128, issue.1, pp.153-179, 1997.
DOI : 10.1006/aima.1997.1634

M. Meyer, Convex bodies with minimal volume product in ?2, Monatshefte f???r Mathematik, vol.55, issue.4, pp.297-301, 1991.
DOI : 10.1007/BF01351770

M. Meyer and E. Werner, On the p-Affine Surface Area, Advances in Mathematics, vol.152, issue.2, pp.288-313, 2000.
DOI : 10.1006/aima.1999.1902

URL : https://hal.archives-ouvertes.fr/hal-00693775

C. M. Petty, Discrete geometry and convexity, pp.113-127, 1982.

R. T. Rockafellar, Convex analysis. Reprint of the 1970 original Princeton Landmarks in Mathematics. Princeton Paperbacks, pp.0-691, 1997.

L. Santaló, An affine invariant for convex bodies of n-dimensional space, Portugaliae Math, pp.155-161, 1949.

G. Sapiro and A. Tannenbaum, On Affine Plane Curve Evolution, Journal of Functional Analysis, vol.119, issue.1, pp.79-120, 1994.
DOI : 10.1006/jfan.1994.1004

URL : http://doi.org/10.1006/jfan.1994.1004

R. Schneider, Convex Bodies: The Brunn-Minkowski theory, 1993.
DOI : 10.1017/CBO9780511526282

F. Schuster and T. Wannerer, $\mathrm{GL}(n)$ contravariant Minkowski valuations, Transactions of the American Mathematical Society, vol.364, issue.2, pp.815-826, 2012.
DOI : 10.1090/S0002-9947-2011-05364-X

F. E. Schuster and M. Weberndorfer, Volume inequalities for asymmetric Wulff shapes, Journal of Differential Geometry, vol.92, issue.1, pp.263-283, 2012.
DOI : 10.4310/jdg/1352297808

C. Schütt and E. Werner, Surface bodies and p-affine surface area, Advances in Mathematics, vol.187, issue.1, pp.98-145, 2004.
DOI : 10.1016/j.aim.2003.07.018

C. Schütt and E. Werner, Polytopes with vertices chosen randomly from the boundary of a convex body, Geometric aspects of Functional Analysis, pp.241-422, 2003.

A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Information and Control, vol.2, issue.2, pp.101-112, 1959.
DOI : 10.1016/S0019-9958(59)90348-1

A. Stancu, The Discrete Planar L0-Minkowski Problem, Advances in Mathematics, vol.167, issue.1, pp.160-174, 2002.
DOI : 10.1006/aima.2001.2040

N. S. Trudinger and X. Wang, The affine Plateau problem, Journal of the American Mathematical Society, vol.18, issue.02, pp.253-289, 2005.
DOI : 10.1090/S0894-0347-05-00475-3

X. Wang, Affine maximal hypersurfaces, Proceedings of the International Congress of Mathematicians, III, pp.221-231, 2002.

E. Werner and D. Ye, New <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msub><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math> affine isoperimetric inequalities, Advances in Mathematics, vol.218, issue.3, pp.762-780, 2008.
DOI : 10.1016/j.aim.2008.02.002

U. , U. , C. Marne-la-vallée, and F. Matthieu, fradelizi@u-pem.fr, olivier.guedon@u-pem