Autour de l'inégalité de Brunn-Minkowski
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 12 (2003) no. 2, pp. 127-178.
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Barthe, Franck. Autour de l'inégalité de Brunn-Minkowski. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 12 (2003) no. 2, pp. 127-178. http://www.numdam.org/item/AFST_2003_6_12_2_127_0/

[1] Artstein ( S. ) , Ball (K. ), Barthe (F.), Naor (A.).- Entropy growth for sums of independent random variables. Submitted, 2002.

[2] Artstein ( S. ) , Ball (K. ), Barthe (F.), Naor (A.). - More on entropy production. Preprint, 2002.

[3] Bakry (D.) , Ledoux ( M.). - Lévy-Gromov isoperimetric inequality for an infinite dimensional diffusion generator. Invent. Math. , 123, p. 259-281 (1996). | MR | Zbl

[4] Ball (K.) , Barthe ( F.), Naor (A.). - Entropy jumps in the presence of a spectral gap. Duke Math. J., 119, p. 41-63 (2003). | MR | Zbl

[5] Ball (K.M. ).- Cube slicing in R×. Proc. Amer. Math. Soc., 97, p. 465-473 (1986). | MR | Zbl

[6] Ball (K.M. ). - Logarithmically concave functions and sections of convex sets in R×. Studia Math., 88, p. 69-84 (1988). | MR | Zbl

[7] Ball (K.M. ). - Volumes of sections of cubes and related problems . In J. Lindenstrauss and V. D. Milman, editors, Israel seminar on Geometric Aspects of Functional Analysis, number 1376 in Lectures Notes in Math. Springer-Verlag , 1989. | MR | Zbl

[8] Ball (K.M. ).- Volume ratio and a reverse isoperimetric inequality. J. London Math. Soc., 44(2), p. 351-359 (1991). | MR | Zbl

[9] Ball (K.M. ). - Mahler's conjecture and wavelets . Discrete Comput. Geom., 13(3-4), p. 271-277 (1995). | MR | Zbl

[10] Ball (K.M. ). - Some remarks on the geometry of convex sets. In Geometric Aspects of Functional Analysis, number 1317 in LMN, p. 224-231. Springer (1998). | MR | Zbl

[11] Barron (A. ) , Johnson ( O.).- Fisher information inequalities and the central limit theorem. Preprint, arXiv, math. PR/0111020. | MR | Zbl

[12] Barron (A.R. ).- Entropy and the central limit theorem . Ann. Probab., 14, p. 336-342 (1986). | MR | Zbl

[13] Barthe (F. ).- Mesures unimodales et sections des boules Bnp. C. R. Acad. Sci. Paris Sér. I Math. , 321, p. 865-868 (1995). | MR | Zbl

[14] Barthe (F. ). - Inégalités de Brascamp-Lieb et convexité . C. R. Acad. Sci. Paris Sér. I Math., 324, p. 885-888 (1997). | MR | Zbl

[15] Barthe (F. ). - Inégalités fonctionnelles et géométriques obtenues par transport des mesures. Thèse de Doctorat, Université de Marne-la-Vallée, 1997.

[16] Barthe (F. ). - An extremal property of the mean width of the simplex . Math. Ann., 310, p. 685-693 (1998). | MR | Zbl

[17] Barthe (F. ). - On a reverse form of the Brascamp-Lieb inequality. Invent. Math., 134, p. 335-361 (1998). | MR | Zbl

[18] Barthe (F. ). - Optimal Young's inequality and its converse, a simple proof. Geom. Funct. Anal., 8, p. 234-242 (1998). | MR | Zbl

[19] Barthe (F. ). - Restricted Prékopa-Leindler inequality . Pacific J. Math., 189, p. 211-222 (1999). | MR | Zbl

[20] Barthe (F. ).- Extremal properties of central half-spaces for product measures. J. Funct. Anal., 182, p. 81-107 (2001). | MR | Zbl

[21] Barthe (F. ). - An isoperimetric result for the Gaussian measure and unconditional sets. Bull. London Math. Soc., 33, p. 408-416 (2001). | MR | Zbl

[22] Barthe (F. ). - Levels of concentration between exponential and Gaussian. Ann. Fac. Sci. Toulouse, 10(3), p. 393-404 (2001). | EuDML | Numdam | MR | Zbl

[23] Barthe (F. ). - Infinite dimensional isoperimetric inequalities in product spaces with the uniform distance. Submitted, 2002 .

[24] Barthe (F. ).- Log-concave and spherical models in isoperimetry. Geom. Funct. Anal., 12, p. 32-55 (2002). | MR | Zbl

[25] Barthe (F. ), Cordero-Erausquin (D.), Fradelizi (M.). - Shift inequalities of Gaussian type and norms of barycenters. Studia. Math. , 146(3), p. 245-259 (2001). | EuDML | MR | Zbl

[26] Barthe (F. ), Csornyei (M.), Naor (A.). - A note on simultaneous polar and Cartesian decomposition. Geometric Aspects of Functional Analysis, to appear. | MR | Zbl

[27] Barthe (F. ), Fradelizi (M.), Maurey (B.). - A short solution to the Busemann-Petty problem. Positivity, 3, p. 95-100 (1999). | MR | Zbl

[28] Barthe (F. ), Koldobsky (A.).- Extremal slabs in the cube and the Laplace transform. Adv. Math., 174, p. 89-114 (2003). | MR | Zbl

[29] Barthe (F. ), Maurey (B.). - Somes remarks on isoperimetry of Gaussian type. Ann. Inst. H. Poincaré, Probabilités et Statistiques, 36(4), p. 419-434 (2000). | EuDML | Numdam | MR | Zbl

[30] Barthe (F. ), Naor ( A.). - Hyperplane projections of the unit ball of lnp. Discrete Comput. Geom, 27(2), p. 215-226 (2002). | MR | Zbl

[31] Beckner ( W.). - Inequalities in Fourier analysis . Ann. of Math., 102, p. 159-182 (1975). | MR | Zbl

[32] Bobkov (S.G. ). - Extremal properties of half-spaces for log-concave distributions. Ann. Probab., 24(1), p. 35-48 (1996). | MR | Zbl

[33] Bobkov (S.G. ). - A functional form of the isoperimetric inequality for the Gaussian measure. J. Funct. Anal, 135, p. 39-49 (1996). | MR | Zbl

[34] Bobkov (S.G. ).- An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probab., 25(1), p. 206-214 (1997). | MR | Zbl

[35] Bobkov (S.G. ). - Isoperimetric problem for uniform enlargement . Studia Math., 123(1), p. 81-95 (1997). | EuDML | MR | Zbl

[36] Bobkov (S.G. ). - Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab., 27(4), p. 1903-1921 (1999 ). | MR | Zbl

[37] Bobkov (S.G. ). - The size of singular component and shift inequalities. Ann. Probab., 27(1), p. 416-431 (1999). | MR | Zbl

[38] Bobkov (S.G. ), Houdré (C.). - Characterization of Gaussian measures in terms of the isoperimetric property of half-spaces. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 228, p. 31-38 (1996). (Russian). | Zbl

[39] Bobkov (S.G. ) , Houdré ( C.). - Isoperimetric constants for product probability measures. Ann. Probab., 25(1), p. 184-205 (1997). | MR | Zbl

[40] Bobkov (S.G. ) , Houdré ( C.). - Weak dimension-free concentration of measure . Bernoulli, 6(4), p. 621-632 (2000). | MR | Zbl

[41] Bobkov (S.G. ), Ledoux (M.). - From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal., 10(5), p. 1028-1052 (2000). | MR | Zbl

[42] Bollobás ( B.), Leader (I.).- Edge-isoperimetric inequalities in the grid. Combinatorica, 11, p. 299-314 (1991). | MR | Zbl

[43] Borell (C. ).- The Brunn-Minkowski inequality in Gauss space. Invent. Math., 30, p. 207-216 (1975). | EuDML | MR | Zbl

[44] Borell (C. ). - Convex functions in d-space. Period. Math. Hungar., 6, p. 111-136 (1975). | MR | Zbl

[45] Bourgain ( J.). - On the Busemann-Petty problem for perturbations of the ball. Geom. Funct. Anal., 1, p. 1-13 (1991). | EuDML | MR | Zbl

[46] Bourgain ( J. ). - On the distribution of polynomials on high dimensional convex sets. In Geometric Aspects of Functional Analysis, number 1469 in Lecture Notes in Math , p. 127-137. Sringer-Verlag , 1991. | MR | Zbl

[47] Bourgain ( J.), Milman (V.D.).- New volume ratio properties for convex symmetric bodies in R×. Invent. Math., 88, p. 319-340 (1987). | EuDML | MR | Zbl

[48] Brascamp ( H.J.), Lieb (E.H.). - Best constants in Young's inequality, its converse and its generalization to more than three functions . Adv. Math., 20, p. 151-173 (1976). | MR | Zbl

[49] Brascamp ( H.J.), Lieb (E.H.). - On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log-concave functions, and with applications to the diffusion equation. J. Funct. Anal., 22, p. 366-389 (1976). | MR | Zbl

[50] Brenier ( Y.).- Décomposition polaire et réarrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris Sér. I Math., 305, p. 805-808 (1987). | MR | Zbl

[51] Brenier ( Y.). - Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math, 44 (1991). | MR | Zbl

[52] Busemann ( H.). - A theorem on convex bodies of Brunn-Minkowski type. Amer. J. Math., 71, p. 743-762 (1949). | MR | Zbl

[53] Caetano ( A.M.). - Weyl numbers in sequence spaces and sections of unit balls. J. Funct. Anal., 106, p. 1-17 (1992). | MR | Zbl

[54] Caffarelli ( L.). - The regularity of mappings with a convex potential. J. Amer. Math. Soc., 4, p. 99-104 (1992). | MR | Zbl

[55] Capitaine ( M.), Hsu (E.P.), Ledoux (M.). - Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces. Elect. Comm. in Probab., 2 (1997). | EuDML | MR | Zbl

[56] Carlen (E.A. ), Soffer (A.). - Entropy production by block variable summation and central limit theorem. Commun. Math. Phys., 140(2), p. 339-371 (1991). | MR | Zbl

[57] Chafaï (D. ), Ledoux (M.). - Méthodes fonctionnelles pour des grandes déviations quasi-gaussiennes. C. R. Acad. Sci. Paris Sér. I Math., 329(6), p. 523-526 (1999). | MR | Zbl

[58] Csiszar (I. ). - Informationstheoretische Konvergenzbegriffe im Raum der Wahrscheinlichkeitsverteilungen. Publications of the Mathematical Institute, Hungarian Academy of Sciences , VII, Series A, p. 137-157 (1962). | MR | Zbl

[59] Ehrhard ( A.). - Symétrisation dans l'espace de Gauss. Math. Scand., 53, p. 281-301 (1983). | EuDML | MR | Zbl

[60] Gardner ( R.J.). - Intersection bodies and the Busemann-Petty problem. Trans. Amer. Math. Soc., 342, p. 435-445 (1994). | MR | Zbl

[61] Gardner (R.J. ). - A positive answer to the Busemann-Petty problem in three dimensions. Ann. of Math., 140, p. 435-447 (1994). | MR | Zbl

[62] Gardner (R.J. ).- Geometric Tomography. Cambridge University Press, New York, 1995. | MR | Zbl

[63] Gardner ( R.J.). - The Brunn-Minkowski inequality . Bull. Amer. Math. Soc. (N.S. ), 3, p. 355-405 (2002). | MR | Zbl

[64] Gardner ( R.J.), Koldobsky (A.), Schlumprecht (Th.). - An analytic solution to the Busemann-Petty problem on section of convex bodies. Ann. of Math. (2), 149(2), p. 691-703 (1999). | EuDML | MR | Zbl

[65] Giannopoulos ( A.). - A note on a problem of H. Busemann and C. M. Petty concerning sections of symmetric convex bobies. Mathematika, 37, p. 239-244 (1990). | MR | Zbl

[66] Gordon (Y. ), Meyer (M.), Reisner (S.).- Zonoids with minimal volume-product. a new proof. Proc. Amer. Math. Soc. , 104, p. 273-276 (1988). | MR | Zbl

[67] Gromov (M. ). - Paul Lévy's isoperimetric inequality . Preprint I.H.E.S., 1980.

[68] Gross (L. ). - Logarithmic Sobolev inequalities. Amer. J. Math., 97, p. 1061-1083 (1975 ). | MR | Zbl

[69] Hadwiger ( H.).- Gitterperiodische Punktmengen und Isoperimetrie. Monatsh. Math., 76, p. 410-418 (1972). | EuDML | MR | Zbl

[70] Hadwiger ( H.), Ohmann (D.).- Brunn-Minkowskischer Satz und Isoperimetrie. Math. Zeit., 66, p. 1-8 (1956). | EuDML | MR | Zbl

[71] Hensley ( D.). - Slicing the cube in Rn and probability . Proc. Amer. Math. Soc., 73(1), p. 95-100 (1979). | MR | Zbl

[72] Hensley ( D.). - Slicing convex bodies - bounds for slice area in terms of body's covariance. Proc. Amer. Math. Soc. , 79(4), p. 619-625 (1980). | MR | Zbl

[73] Henstock ( R.), Macbeath (A.H.). - On the measure of sum sets. (I) the theorems of Brunn, Minkowski and Lusternik. Proc. London Math. Soc., 3, p. 182-194 (1953). | MR | Zbl

[74] John (F.). - Extremum problems with inequalities as subsidiary conditions . In Courant Anniversary Volume, p. 187-204, New York, 1948. Interscience. | MR | Zbl

[75] Knothe (H. ). - Contributions to the theory of convex bodies. Michigan Math. J., 4, p. 39-52 (1957). | MR | Zbl

[76] Koldobsky ( A.). - Intersection bodies and the Busemann-Petty problem. C. R. Acad. Sci. Paris Sér. I Math., 325, p. 1181-1186 (1997). | MR | Zbl

[77] Koldobsky ( A.).- An application of the Fourier transform to sections of star bodies. Israel J. Math., 106, p. 157-164 (1998). | MR | Zbl

[78] Koldobsky ( A.). - Intersection bodies in R4. Adv. Math., 136(1), p. 1-14 (1998). | MR | Zbl

[79] Kuelbs (J. ) , Li ( W.V. ). - Some shift inequalities for Gaussian measures . In High dimensional probability (Oberwolfach, 1996), Progr. Probab, p. 233-243, Basel, 1998 . Birkhäuser. | MR | Zbl

[80] Kullback ( S.). - A lower bound for discrimination information in terms of variation. IEEE Trans. Info. Theory , 4, p. 126-127 (1967).

[81] Kwapien ( S.), Pycia (M.), Schachermayer (W.).- A proof of a conjecture of Bobkov and Houdré. Elect. Comm. in Probab. , 1, p. 7-10 (1996). | EuDML | MR | Zbl

[82] Larman (D.G. ), Rogers (C.A.). - The existence of a centrally symmetric convex body with central sections that are unexpectedly small. Mathematika, 22, p. 164-175 (1975). | MR | Zbl

[83] Latała ( R.), Oleszkiewicz (K.). - Between Sobolev and Poincaré. In Geometric aspects of functional analysis, number 1745 in Lecture Notes in Math, p. 147-168, Berlin, 2000. Springer. | MR | Zbl

[84] Ledoux (M. ). - The concentration of measure phenomenon, volume 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2001 . | MR | Zbl

[85] Leindler ( L. ). - On a certain converse of Hôlder's inequality. II . Acta Sci. Math. Szeged, 33, p. 217-223 (1972). | Zbl

[86] Lévy (P.). - Problèmes concrets d'analyse fonctionnelle. Gauthiers-Villars, Paris, 1951. | Zbl

[87] Lieb (E.H. ). - Proof of an entropy conjecture of Wehrl. Commun. math. Phys., 62, p. 35-41 (1978). | MR | Zbl

[88] Lieb (E.H. ). - Gaussian kernels have only gaussian maximizers . Invent. Math., 102, p. 179-208 (1990). | EuDML | MR | Zbl

[89] Linnik (Ju.V. ). - An information theoretic proof of the central limit theorem with lindeberg conditions. Theory Probab. Appl., 4, p. 288-299 (1959). | MR | Zbl

[90] Lutwak (E. ). - Intersection bodies and dual mixed volumes . Adv. Math., 71, p. 232-261 (1988). | MR | Zbl

[91] Maurey (B. ). - Some deviation inequalities. Geom. Funct. Anal., 1(2), p. 188-197 (1991). | EuDML | MR | Zbl

[92] Mccann (R.J. ). - A Convexity Theory for Interacting Gases and Equilibrium Crystals. PhD thesis, Princeton University , 1994.

[93] Mccann (R.J. ). - A convexity principle for interacting gases . Adv. Math., 128, p. 153-179 (1997). | MR | Zbl

[94] Meyer (M. ), Pajor (A.).- Sections of the unit ball of lpn. J. Funct. Anal., 80, p. 109-123 (1988). | MR | Zbl

[95] Milman (V. ) , Schechtman ( G.). - Asymptotic Theory of Finite Dimensional Normed Spaces. Number 1200 in Lecture Notes in Math Springer Verlag, 1986. | MR | Zbl

[96] Milman (V.D. ) , Pajor ( A.). - Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. In Geometric Aspects of Functional Analysis, number 1376 in LMN, p. 64-104. Springer, ( 1989). | MR | Zbl

[97] Oleszkiewicz ( K.). - On certain characterization of normal distribution. Statist. Probab. Lett., 33(3), p. 277-280 (1997). | MR | Zbl

[98] Papadimitrakis ( M.). - On the Busemann-Petty problem about convex, centrally symmetric bodies in Rn. Mathematika, 39, p. 258-266 (1992). | MR | Zbl

[99] Petty (C.M. ).- Projection bodies. In Proc. Colloquium Convexity, pages 234-241, Copenhagen, 1965. Kobenhavns Univ. Math. Inst. | MR | Zbl

[100] Pinsker (M.S. ). - Information and information stability of random variables and processes. Holden-Day, San Francisco, 1964. | MR | Zbl

[101] Pisier (A. ). - The volume of convex bodies and Banach space geometry, volume 94 of Cambridge Tracts in Mathematics. - Cambridge University Press, Cambridge , 1989. | MR | Zbl

[102] Prékopa ( A.). - On logarithmic concave measures and functions . Acta Scient. Math., 34, p. 335-343 (1973). | MR | Zbl

[103] Reisner ( S.). - Random polytopes and the volume product of symmetric convex bodies. Math. Scand., 57(2), p. 386-392 (1985). | MR | Zbl

[104] Rinott ( Y.). - On convexity of measures. Ann. Probab., 4, p. 1020-1026 (1976). | MR | Zbl

[105] Ros (A.). - The isoperimetric problem. http, //www.ugr.es/aros/#Isoperimetric, 2001.

[106] Saint-Raymond ( J.). - Sur le volume des corps convexes symétriques . In Séminaire d'initiation à l'analyse. 80/81. Exp. 11 , Paris, 1981. Publ. Math. Univ. Pierre et Marie Curie, Univ. Paris VI. | MR | Zbl

[107] Schechtman ( G. ), Schmuckenschläger (M.).- A concentration inequality for harmonic measures on the sphere. In Geometric aspects of functional analysis (Israel, 1992-1994, number 77 in Oper. Theory Adv. Appl., pages 255-273, Basel, 1995. Birkhauser. | MR | Zbl

[108] Scheffer ( G. ). - Isopérimétrie fonctionnelle dimensionnelle en courbure positive. C. R. Acad. Sci. Paris, Sér. I Math. , 331, p. 251-254 (2001). | MR | Zbl

[109] Schmidt ( E.). - Die Brunn-Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und nichteuklidischen Geometrie I, II. Math. Nachr., 1, p. 81-157 (1948). 2, p. 171-244 (1949). | MR | Zbl

[110] Schmuckenschläger ( M.).- A concentration of measure phenomenon on uniformly convex bodies. In Geometric Aspects of Functional Analysis (Israel 1992-94), number 77 in Oper. Theory Adv. Appl., p. 275-287. Birkhäuser, 1995. | MR | Zbl

[111] Schmuckenschläger ( M.). - An extremal property of the regular simplex . In Convex geometric analysis (Berkeley, CA, 1996), volume 34 of Math. Sci. Res. Inst. Publ., p. 199-202, Cambridge, 1999. Cambridge Univ. Press. | MR | Zbl

[112] Schneider ( R.). - Zu einem Problem von Shephard über die Projektionen konvexer Kôrper. Math. Z., 101, p. 71-82 (1967). | Zbl

[113] Schneider ( R. ). - Convex bodies, the Brunn-Minkowski theory, volume 44 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1993. | MR | Zbl

[114] Shannon (C.E. ), Weaver (W.).- The mathematical theory of communication. University of Illinois Press, Urbana, IL, 1949 . | MR | Zbl

[115] Stam (A.J. ). - Some inequalities satisfied by the quantities of information of Fisher and Shannon. - Info. Control, 2, p. 101-112 (1959). | MR | Zbl

[116] Sudakov ( V.N.), Tsirel'Son (B.S.). - Extremal propreties of half-spaces for spherically invariant measures. J. Soviet Math., 9, 9-18, 1978. | Zbl

Translated from Zap. Nauchn. Sem. Leningrad. Otdel. Math. Inst. Steklova. 41, p. 14-24 (1974).

[117] Szarek ( S.), Voiculescu (D.). - Volumes of restricted Minkowski sums and the free analogue of the entropy power inequality. Commun. Math. Phys., 178(3), p. 563-570 (1996). | MR | Zbl

[118] Szarek ( S.), Voiculescu (D.).- Shannon's entropy power inequality via restricted Minkowski sums. In Geometric aspects of functional analysis, volume 1745 of Lecture Notes in Math., p. 257-262. Springer, 2000. | MR | Zbl

[119] Uhrin (B. ). - Curvilinear extensions of the Brunn-Minkowski-Lusternik inequality. Adv. Math., 109(2), p. 288-312 (1994). | MR | Zbl

[120] Vaaler ( J.D.). - A geometric inequality with applications to linear forms. Pacific J. Math., 83, p. 543-553 (1979). | MR | Zbl

[121] Zhang (G. ). - Centered bodies and dual mixed volumes . Trans. Amer. Soc., 345, p. 777-801 (1994). | MR | Zbl

[122] Zhang (G. ). - A positive answer to the Busemann-Petty problem in four dimensions. Ann. of Math. (2), 149(2), p. 535-543 (1999). | MR | Zbl