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The Brunn-Minkowski inequality. (English) Zbl 1019.26008

This is a basic and high quality survey on the subject related to the isoperimetric inequality. As the author writes: “This guide explains the relationship between Brunn-Minkowski inequality (B-M-I) and other inequalities in geometry and analysis, and some applications.”
This work can be considered as the up-to-date version of the excellent survey due to Osserman (1978). I do believe that the best is to read this extensive survey from \(A\) to \(Z\). To illustrate the wealth of the paper we shall recall some sentences of the introduction (occasionally in slightly modified form): “The fundamental geometric content of the B-M-I makes it a cornerstone of the Brunn-Minkowski theory, a beautiful and powerful apparatus for conquering all sorts of problems involving metric quantities such as volume and surface area.
By the mid-twentieth century, however, when Lusternik, Hadwiger and Ohmann, and Henstock and Macbeath had established a satisfactory generalization of B-M-I and its equality condition to Lebesgue measurable sets, the inequality had begun its move into the realm of analysis. The last twenty years have seen the B-M-I consolidate its role as an analytical tool. In an integral version of the B-M-I often called the Prékopa-Leindler inequality, a reverse form of Hölder’s inequality, the geometry seems to have evaporated. Largely through the efforts of Brascamp and Lieb, this inequality can be viewed as a special case of a sharp reverse form of Young’s inequality for convolution norms. A remarkable sharp inequality proved by Barthe takes us up to the present time. The modern viewpoint entails an interaction between analysis and convex geometry.
Several applications are discussed at some length. Section 6 explains why the B-M-I can be applied to the Wulff shape of crystals. McCann’s work on gases, in which the B-M-I appears is introduced in Section 8, along with a crucial idea called transport of mass that was also used by Barthe in his proof of the Brascamp-Lieb and Barthe inequalities. Section 9 explains that the Prékopa-Leindler inequality can be used to show that a convolution of log-concave functions is log concave, and an application to diffusion equations is outlined. The Prékopa-Leindler inequality can also be applied to prove that certain measures are log concave.
The Borell-Brascamp-Lieb inequality, an extension of the Prékopa-Leindler inequality introduced in section 10, is very useful in probability theory and statistics. Such applications are treated in Section 11, along with related consequences of Anderson’s theorem on multivariate unimodality, the proof of which employs the B-M-I. The entropy power inequality of information theory has a form similar to that of the B-M-I. Section 14 elaborates on this and related matters, such as Fisher information, uncertainty principles, and logarithmic Sobolev inequalities. In Section 16, we come full circle with applications to geometry. Keith Ball started these rolling with his elegant application of the Brascamp-Lieb inequality. Milman’s reverse B-M-I features prominently in the local theory of Banach spaces.
Section 12 brings versions of the B-M-I in the sphere, hyperbolic space, Minkowski spacetime, and Gauss space, and a Riemannian version of the Borell-Brascamp-Lieb inequality, obtained very recently by Cordero-Erausquin, McCann, and Schmuckenschläger. In Section 17 a remarkable link with algebraic geometry is sketched: Khovanskii and Teissier independently discovered that the Aleksandrov-Fenchel inequality can be deduced from the Hodge index theorem. Analogues and variants of the B-M-I include Borell’s inequality for capacity, employed in the recent solution of the Minkowski problem for capacity; a discrete B-M-I due to the author and Gronchi, closely related to a rich area of discrete mathematics, combinatorics, and graph theory concerning discrete isoperimetric inequalities; and some other inequalities originating in Busemann’s theorem, motivated by his theory of area in Finsler spaces and used in Minkowski-geometry and geometric tomography.”
The author raises some essential problems, too.

MSC:

26D15 Inequalities for sums, series and integrals
52A40 Inequalities and extremum problems involving convexity in convex geometry
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[1] S. Alesker, S. Dar, and V. Milman, A remarkable measure preserving diffeomorphism between two convex bodies in \?\(^{n}\), Geom. Dedicata 74 (1999), no. 2, 201 – 212. · Zbl 0927.52007
[2] T. W. Anderson, The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities, Proc. Amer. Math. Soc. 6 (1955), 170-176. · Zbl 0066.37402
[3] Ben Andrews, Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138 (1999), no. 1, 151 – 161. · Zbl 0936.35080
[4] Juan Arias-de-Reyna, Keith Ball, and Rafael Villa, Concentration of the distance in finite-dimensional normed spaces, Mathematika 45 (1998), no. 2, 245 – 252. · Zbl 0956.46009
[5] Hyoungsick Bahn and Paul Ehrlich, A Brunn-Minkowski type theorem on the Minkowski spacetime, Canad. J. Math. 51 (1999), no. 3, 449 – 469. · Zbl 0978.53042
[6] R. Baierlein, Atoms and Information Theory, W. H. Freeman and Company, San Francisco, 1971.
[7] Ilya J. Bakelman, Convex analysis and nonlinear geometric elliptic equations, Springer-Verlag, Berlin, 1994. With an obituary for the author by William Rundell; Edited by Steven D. Taliaferro. · Zbl 0815.35001
[8] Keith Ball, Logarithmically concave functions and sections of convex sets in \?\(^{n}\), Studia Math. 88 (1988), no. 1, 69 – 84. · Zbl 0642.52011
[9] Keith Ball, Volumes of sections of cubes and related problems, Geometric aspects of functional analysis (1987 – 88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 251 – 260.
[10] Keith Ball, Shadows of convex bodies, Trans. Amer. Math. Soc. 327 (1991), no. 2, 891 – 901. · Zbl 0746.52007
[11] Keith Ball, Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. (2) 44 (1991), no. 2, 351 – 359. · Zbl 0694.46010
[12] Keith Ball, An elementary introduction to modern convex geometry, Flavors of geometry, Math. Sci. Res. Inst. Publ., vol. 31, Cambridge Univ. Press, Cambridge, 1997, pp. 1 – 58. · Zbl 0832.11018
[13] Franck Barthe, Inégalités de Brascamp-Lieb et convexité, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 8, 885 – 888 (French, with English and French summaries). · Zbl 0904.26011
[14] -, Inégalités Fonctionelles et Géométriques Obtenues par Transport des Mesures, Ph.D. thesis, Université de Marne-la-Vallée, Paris, 1997.
[15] A. Volčič, Generalized Hammer’s X-ray problem, Atti Sem. Mat. Fis. Univ. Modena 46 (1998), no. suppl., 393 – 400. Dedicated to Prof. C. Vinti (Italian) (Perugia, 1996). · Zbl 0919.52009
[16] Franck Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math. 134 (1998), no. 2, 335 – 361. · Zbl 0901.26010
[17] F. Barthe, Optimal Young’s inequality and its converse: a simple proof, Geom. Funct. Anal. 8 (1998), no. 2, 234 – 242. · Zbl 0902.26009
[18] Franck Barthe, Restricted Prékopa-Leindler inequality, Pacific J. Math. 189 (1999), no. 2, 211 – 222. · Zbl 0935.26012
[19] Edwin F. Beckenbach and Richard Bellman, Inequalities, Second revised printing. Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Band 30, Springer-Verlag, New York, Inc., 1965. · Zbl 0126.28002
[20] William Beckner, Inequalities in Fourier analysis, Ann. of Math. (2) 102 (1975), no. 1, 159 – 182. · Zbl 0338.42017
[21] F. Behrend, Über die kleinste umbeschriebene und die grösste einbeschriebene Ellipse eines konvexen Bereichs, Math. Ann. 115 (1938), 379-411. · Zbl 0018.17502
[22] G. Bianchi, Determining convex bodies with piecewise \({C}^2\) boundary from their covariogram, preprint.
[23] Nelson M. Blachman, The convolution inequality for entropy powers, IEEE Trans. Information Theory IT-11 (1965), 267 – 271. · Zbl 0134.37401
[24] S. G. Bobkov and M. Ledoux, From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities, Geom. Funct. Anal. 10 (2000), 1028-1052. CMP 2001:05 · Zbl 0969.26019
[25] Béla Bollobás and Imre Leader, Sums in the grid, Discrete Math. 162 (1996), no. 1-3, 31 – 48. · Zbl 0872.11007
[26] Christer Borell, The Brunn-Minkowski inequality in Gauss space, Invent. Math. 30 (1975), no. 2, 207 – 216. · Zbl 0292.60004
[27] C. Borell, Convex set functions in \?-space, Period. Math. Hungar. 6 (1975), no. 2, 111 – 136. · Zbl 0274.28009
[28] Christer Borell, Capacitary inequalities of the Brunn-Minkowski type, Math. Ann. 263 (1983), no. 2, 179 – 184. · Zbl 0546.31001
[29] Christer Borell, Geometric properties of some familiar diffusions in \?\(^{n}\), Ann. Probab. 21 (1993), no. 1, 482 – 489. · Zbl 0776.35024
[30] Christer Borell, Geometric inequalities in option pricing, Convex geometric analysis (Berkeley, CA, 1996) Math. Sci. Res. Inst. Publ., vol. 34, Cambridge Univ. Press, Cambridge, 1999, pp. 29 – 51. · Zbl 1050.91036
[31] Christer Borell, Diffusion equations and geometric inequalities, Potential Anal. 12 (2000), no. 1, 49 – 71. · Zbl 0976.60065
[32] H. J. Brascamp and E. H. Lieb, Some inequalities for Gaussian measures and the long-range order of one-dimensional plasma, Functional Integration and Its Applications, ed. by A. M. Arthurs, Clarendon Press, Oxford, 1975, pp. 1-14. · Zbl 0348.26011
[33] Herm Jan Brascamp and Elliott H. Lieb, Best constants in Young’s inequality, its converse, and its generalization to more than three functions, Advances in Math. 20 (1976), no. 2, 151 – 173. · Zbl 0339.26020
[34] Herm Jan Brascamp and Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), no. 4, 366 – 389. · Zbl 0334.26009
[35] V. V. Buldygin and A. B. Kharazishvili, Geometric aspects of probability theory and mathematical statistics, Mathematics and its Applications, vol. 514, Kluwer Academic Publishers, Dordrecht, 2000. Translated from the 1985 Russian original, The Brunn-Minkowski inequality and its applications [”Naukova Dumka”, Kiev, 1985; MR0889670 (89b:52019)], by Kharazishvili and revised by the authors. · Zbl 0968.60002
[36] Геометрические неравенства, ”Наука” Ленинград. Отдел., Ленинград, 1980 (Руссиан). · Zbl 0633.53002
[37] H. Busemann, The isoperimetric problem for Minkowski area, Amer. J. Math. 71 (1949), 743-762. · Zbl 0038.10301
[38] Luis A. Caffarelli, David Jerison, and Elliott H. Lieb, On the case of equality in the Brunn-Minkowski inequality for capacity, Adv. Math. 117 (1996), no. 2, 193 – 207. · Zbl 0847.31005
[39] D. Cordero-Erausquin, Some applications of mass transport to Gaussian type inequalities, Arch. Rational Mech. Anal., to appear. · Zbl 0998.60080
[40] Dario Cordero-Erausquin, Inégalité de Prékopa-Leindler sur la sphère, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 9, 789 – 792 (French, with English and French summaries). · Zbl 0945.26026
[41] D. Cordero-Erausquin, R. J. McCann, and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math. 146 (2001), 219-257. · Zbl 1026.58018
[42] Max H. M. Costa and Thomas M. Cover, On the similarity of the entropy power inequality and the Brunn-Minkowski inequality, IEEE Trans. Inform. Theory 30 (1984), no. 6, 837 – 839. · Zbl 0557.94006
[43] S. Dancs and B. Uhrin, On a class of integral inequalities and their measure-theoretic consequences, J. Math. Anal. Appl. 74 (1980), no. 2, 388 – 400. · Zbl 0442.26011
[44] I. Dancs and B. Uhrin, On the conditions of equality in an integral inequality, Publ. Math. Debrecen 29 (1982), no. 1-2, 117 – 132. · Zbl 0525.26012
[45] S. Dar, A Brunn-Minkowski-type inequality, Geom. Dedicata 77 (1999), no. 1, 1 – 9. · Zbl 0938.52008
[46] Somesh Das Gupta, Brunn-Minkowski inequality and its aftermath, J. Multivariate Anal. 10 (1980), no. 3, 296 – 318. · Zbl 0467.26008
[47] Amir Dembo, Thomas M. Cover, and Joy A. Thomas, Information-theoretic inequalities, IEEE Trans. Inform. Theory 37 (1991), no. 6, 1501 – 1518. · Zbl 0741.94001
[48] Sudhakar Dharmadhikari and Kumar Joag-Dev, Unimodality, convexity, and applications, Probability and Mathematical Statistics, Academic Press, Inc., Boston, MA, 1988. · Zbl 0646.62008
[49] Alexander Dinghas, Über eine Klasse superadditiver Mengenfunktionale von Brunn-Minkowski-Lusternikschem Typus, Math. Z. 68 (1957), 111 – 125 (German). · Zbl 0083.38301
[50] Richard M. Dudley, Real analysis and probability, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1989. · Zbl 0686.60001
[51] R. M. Dudley, Metric marginal problems for set-valued or non-measurable variables, Probab. Theory Related Fields 100 (1994), no. 2, 175 – 189. · Zbl 0923.60002
[52] H. G. Eggleston, Convexity, Cambridge Tracts in Mathematics and Mathematical Physics, No. 47, Cambridge University Press, New York, 1958. · Zbl 0086.15302
[53] Antoine Ehrhard, Symétrisation dans l’espace de Gauss, Math. Scand. 53 (1983), no. 2, 281 – 301 (French). · Zbl 0542.60003
[54] Antoine Ehrhard, Éléments extrémaux pour les inégalités de Brunn-Minkowski gaussiennes, Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 2, 149 – 168 (French, with English summary). · Zbl 0595.60020
[55] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. · Zbl 0804.28001
[56] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801
[57] William J. Firey, Polar means of convex bodies and a dual to the Brunn-Minkowski theorem, Canad. J. Math. 13 (1961), 444 – 453. · Zbl 0126.18005
[58] Wm. J. Firey, \?-means of convex bodies, Math. Scand. 10 (1962), 17 – 24. · Zbl 0188.27303
[59] William J. Firey, Shapes of worn stones, Mathematika 21 (1974), 1 – 11. · Zbl 0311.52003
[60] Irene Fonseca, The Wulff theorem revisited, Proc. Roy. Soc. London Ser. A 432 (1991), no. 1884, 125 – 145. · Zbl 0725.49017
[61] Irene Fonseca and Stefan Müller, A uniqueness proof for the Wulff theorem, Proc. Roy. Soc. Edinburgh Sect. A 119 (1991), no. 1-2, 125 – 136. · Zbl 0752.49019
[62] B. Roy Frieden, Physics from Fisher information, Cambridge University Press, Cambridge, 1998. A unification. · Zbl 0998.81512
[63] R. J. Gardner, The Brunn-Minkowski inequality: A survey with proofs, available at http://www.ac.wwu.edu/ gardner. · Zbl 1019.26008
[64] R. J. Gardner, Intersection bodies and the Busemann-Petty problem, Trans. Amer. Math. Soc. 342 (1994), no. 1, 435 – 445. · Zbl 0801.52005
[65] R. J. Gardner, A positive answer to the Busemann-Petty problem in three dimensions, Ann. of Math. (2) 140 (1994), no. 2, 435 – 447. · Zbl 0826.52010
[66] Richard J. Gardner, Geometric tomography, Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, Cambridge, 1995. · Zbl 0864.52001
[67] R. J. Gardner and P. Gronchi, A Brunn-Minkowski inequality for the integer lattice, Trans. Amer. Math. Soc. 353 (2001), 3995-4024. · Zbl 0977.52019
[68] R. J. Gardner, A. Koldobsky, and T. Schlumprecht, An analytic solution to the Busemann-Petty problem on sections of convex bodies, Ann. of Math. (2) 149 (1999), no. 2, 691 – 703. · Zbl 0937.52003
[69] R. J. Gardner and Gaoyong Zhang, Affine inequalities and radial mean bodies, Amer. J. Math. 120 (1998), no. 3, 505 – 528. · Zbl 0908.52001
[70] H. Groemer, Stability of geometric inequalities, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 125 – 150. · Zbl 0789.52001
[71] M. Gromov, Convex sets and Kähler manifolds, Advances in differential geometry and topology, World Sci. Publ., Teaneck, NJ, 1990, pp. 1 – 38. · Zbl 0770.53042
[72] Leonard Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061 – 1083. · Zbl 0318.46049
[73] O.-G. Guleryuz, E. Lutwak, D. Yang, and G. Zhang, Information theoretic inequalities for contoured probability distributions, preprint. · Zbl 1062.94530
[74] H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). · Zbl 0078.35703
[75] H. Hadwiger and D. Ohmann, Brunn-Minkowskischer Satz und Isoperimetrie, Math. Zeit. 66 (1956), 1-8. · Zbl 0071.38001
[76] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1959. · Zbl 0634.26008
[77] R. Henstock and A. M. Macbeath, On the measure of sum sets, I. The theorems of Brunn, Minkowski and Lusternik, Proc. London Math. Soc. 3 (1953), 182-194. · Zbl 0052.18302
[78] David Jerison, A Minkowski problem for electrostatic capacity, Acta Math. 176 (1996), no. 1, 1 – 47. · Zbl 0880.35041
[79] Jeff Kahn and Nathan Linial, Balancing extensions via Brunn-Minkowski, Combinatorica 11 (1991), no. 4, 363 – 368. · Zbl 0735.06004
[80] R. Kannan, L. Lovász, and M. Simonovits, Isoperimetric problems for convex bodies and a localization lemma, Discrete Comput. Geom. 13 (1995), no. 3-4, 541 – 559. · Zbl 0824.52012
[81] S. P. King, website at http://members.home.net/stephenk1/Outlaw/fisherinfo.html.
[82] J. F. C. Kingman and S. J. Taylor, Introduction to measure and probability, Cambridge University Press, London-New York-Ibadan, 1966. · Zbl 0171.38603
[83] H. Knothe, Contributions to the theory of convex bodies, Michigan Math. J. 4 (1957), 39-52. · Zbl 0077.35803
[84] Rafał Latała, A note on the Ehrhard inequality, Studia Math. 118 (1996), no. 2, 169 – 174. · Zbl 0847.60012
[85] M. Ledoux, Concentration of measure and logarithmic Sobolev inequalities, Séminaire de Probabilités, ed. by J. Azéma, M. Émery, M. Ledoux, and M. Yor, Lecture Notes in Mathematics 1709, Springer, Berlin, 1999, pp. 120-216. CMP 2000:16
[86] -, The Concentration of Measure Phenomenon, American Mathematical Society, Providence, RI, 2001. · Zbl 0995.60002
[87] Michel Ledoux and Michel Talagrand, Probability in Banach spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23, Springer-Verlag, Berlin, 1991. Isoperimetry and processes. · Zbl 0748.60004
[88] L. Leindler, On a certain converse of Hölder’s inequality. II, Acta Sci. Math. (Szeged) 33 (1972), 217-223. · Zbl 0245.26011
[89] Elliott H. Lieb, Proof of an entropy conjecture of Wehrl, Comm. Math. Phys. 62 (1978), no. 1, 35 – 41. · Zbl 0385.60089
[90] Elliott H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990), no. 1, 179 – 208. · Zbl 0726.42005
[91] Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. · Zbl 0966.26002
[92] J. Lindenstrauss and V. D. Milman, The local theory of normed spaces and its applications to convexity, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 1149 – 1220. · Zbl 0791.52003
[93] L. Lovász and M. Simonovits, Random walks in a convex body and an improved volume algorithm, Random Structures Algorithms 4 (1993), no. 4, 359 – 412. · Zbl 0788.60087
[94] L. A. Lusternik, Die Brunn-Minkowskische Ungleichung für beliebige messbare Mengen, C. R. (Doklady) Acad. Sci. URSS 8 (1935), 55-58. · Zbl 0012.27203
[95] Erwin Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), no. 2, 531 – 538. · Zbl 0273.52007
[96] Erwin Lutwak, Width-integrals of convex bodies, Proc. Amer. Math. Soc. 53 (1975), no. 2, 435 – 439. · Zbl 0276.52006
[97] Erwin Lutwak, A general isepiphanic inequality, Proc. Amer. Math. Soc. 90 (1984), no. 3, 415 – 421. · Zbl 0534.52011
[98] Erwin Lutwak, Volume of mixed bodies, Trans. Amer. Math. Soc. 294 (1986), no. 2, 487 – 500. · Zbl 0591.52016
[99] Erwin Lutwak, Intersection bodies and dual mixed volumes, Adv. in Math. 71 (1988), no. 2, 232 – 261. · Zbl 0657.52002
[100] Erwin Lutwak, Centroid bodies and dual mixed volumes, Proc. London Math. Soc. (3) 60 (1990), no. 2, 365 – 391. · Zbl 0703.52005
[101] Erwin Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), no. 1, 131 – 150. · Zbl 0788.52007
[102] Paola De Vito, On a class of linear spaces with mutually intersecting long lines, Ricerche Mat. 40 (1991), no. 1, 27 – 32 (Italian, with English summary). · Zbl 0757.51011
[103] Erwin Lutwak, Selected affine isoperimetric inequalities, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 151 – 176. · Zbl 0847.52006
[104] Erwin Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math. 118 (1996), no. 2, 244 – 294. · Zbl 0853.52005
[105] E. Lutwak, D. Yang, and G. Zhang, The Brunn-Minkowski-Firey inequality for non-convex sets, preprint. · Zbl 1252.52006
[106] -, The Cramer-Rao inequality for star bodies, Duke Math. J. 112 (2002), 59-81. · Zbl 1021.52008
[107] -, On the \({L}_p\)-Minkowski problem, preprint.
[108] -, Sharp affine \({L}_p\) Sobolev inequalities, preprint.
[109] Erwin Lutwak, Deane Yang, and Gaoyong Zhang, A new ellipsoid associated with convex bodies, Duke Math. J. 104 (2000), no. 3, 375 – 390. · Zbl 0974.52008
[110] -, \({L}_p\) affine isoperimetric inequalities, J. Diff. Geom. 56 (2000), 111-132.
[111] Erwin Lutwak and Gaoyong Zhang, Blaschke-Santaló inequalities, J. Differential Geom. 47 (1997), no. 1, 1 – 16. · Zbl 0906.52003
[112] Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. · Zbl 0819.28004
[113] B. Maurey, Some deviation inequalities, Geom. Funct. Anal. 1 (1991), no. 2, 188 – 197. · Zbl 0756.60018
[114] R. J. McCann, A Convexity Theory for Interacting Gases and Equilibrium Crystals, Ph.D. dissertation, Princeton University, 1994.
[115] Robert J. McCann, A convexity principle for interacting gases, Adv. Math. 128 (1997), no. 1, 153 – 179. · Zbl 0901.49012
[116] Robert J. McCann, Equilibrium shapes for planar crystals in an external field, Comm. Math. Phys. 195 (1998), no. 3, 699 – 723. · Zbl 0936.74029
[117] Peter McMullen, New combinations of convex sets, Geom. Dedicata 78 (1999), no. 1, 1 – 19. · Zbl 0952.52004
[118] J. Mecke and A. Schwella, Inequalities in the sense of Brunn-Minkowski, Vitale for random convex bodies, preprint.
[119] Mathieu Meyer, Maximal hyperplane sections of convex bodies, Mathematika 46 (1999), no. 1, 131 – 136. · Zbl 0988.52019
[120] V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed \?-dimensional space, Geometric aspects of functional analysis (1987 – 88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 64 – 104. · Zbl 0679.46012
[121] Vitali D. Milman and Gideon Schechtman, Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics, vol. 1200, Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov. · Zbl 0606.46013
[122] Melvyn B. Nathanson, Additive number theory, Graduate Texts in Mathematics, vol. 165, Springer-Verlag, New York, 1996. Inverse problems and the geometry of sumsets. · Zbl 0859.11003
[123] Andrei Okounkov, Brunn-Minkowski inequality for multiplicities, Invent. Math. 125 (1996), no. 3, 405 – 411. · Zbl 0893.52004
[124] Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182 – 1238. · Zbl 0411.52006
[125] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations 26 (2001), 101-174. · Zbl 0984.35089
[126] F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000), no. 2, 361 – 400. · Zbl 0985.58019
[127] Gilles Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989. · Zbl 0698.46008
[128] András Prékopa, Logarithmic concave measures with application to stochastic programming, Acta Sci. Math. (Szeged) 32 (1971), 301 – 316. · Zbl 0235.90044
[129] András Prékopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged) 34 (1973), 335 – 343.
[130] András Prékopa, Stochastic programming, Mathematics and its Applications, vol. 324, Kluwer Academic Publishers Group, Dordrecht, 1995. · Zbl 0863.90116
[131] Yosef Rinott, On convexity of measures, Ann. Probability 4 (1976), no. 6, 1020 – 1026. · Zbl 0347.60003
[132] Imre Z. Ruzsa, The Brunn-Minkowski inequality and nonconvex sets, Geom. Dedicata 67 (1997), no. 3, 337 – 348. · Zbl 0888.52011
[133] Michel Schmitt, On two inverse problems in mathematical morphology, Mathematical morphology in image processing, Opt. Engrg., vol. 34, Dekker, New York, 1993, pp. 151 – 169.
[134] Michael Schmuckenschläger, An extremal property of the regular simplex, Convex geometric analysis (Berkeley, CA, 1996) Math. Sci. Res. Inst. Publ., vol. 34, Cambridge Univ. Press, Cambridge, 1999, pp. 199 – 202. · Zbl 0933.52010
[135] Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. · Zbl 0798.52001
[136] J. Serra, Image analysis and mathematical morphology, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1984. English version revised by Noel Cressie.
[137] C. E. Shannon, A mathematical theory of communication, Bell System Tech. J. 27 (1948), 623-656. Can be downloaded at http://www.math.washington.edu/  hillman/Entropy/infcode.html. · Zbl 1154.94303
[138] W. Sierpinski, Sur la question de la mesurabilité de la base de M. Hamel, Fund. Math. 1 (1920), 105-111. · JFM 47.0180.03
[139] A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Information and Control 2 (1959), 101 – 112. · Zbl 0085.34701
[140] A. Stancu, The discrete planar \(L_0\)-Minkowski problem, Adv. Math., to appear. · Zbl 1005.52002
[141] D. Stoyan, W. S. Kendall, and J. Mecke, Stochastic geometry and its applications, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1987. With a foreword by D. G. Kendall. Dietrich Stoyan, Wilfrid S. Kendall, and Joseph Mecke, Stochastic geometry and its applications, Mathematische Lehrbücher und Monographien, II. Abteilung: Mathematische Monographien [Mathematical Textbooks and Monographs, Part II: Mathematical Monographs], vol. 69, Akademie-Verlag, Berlin, 1987. With a foreword by David Kendall. · Zbl 0838.60002
[142] V. N. Sudakov and B. S. Cirel\(^{\prime}\)son, Extremal properties of half-spaces for spherically invariant measures, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 (1974), 14 – 24, 165 (Russian). Problems in the theory of probability distributions, II.
[143] Stanislaw J. Szarek and Dan Voiculescu, Volumes of restricted Minkowski sums and the free analogue of the entropy power inequality, Comm. Math. Phys. 178 (1996), no. 3, 563 – 570. · Zbl 0863.46042
[144] Jean E. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc. 84 (1978), no. 4, 568 – 588. · Zbl 0392.49022
[145] Yung Liang Tong, Probability inequalities in multivariate distributions, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. Probabilities and Mathematical Statistics.
[146] Neil S. Trudinger, Isoperimetric inequalities for quermassintegrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), no. 4, 411 – 425 (English, with English and French summaries). · Zbl 0859.52001
[147] B. Uhrin, Extensions and sharpenings of Brunn-Minkowski and Bonnesen inequalities, Intuitive geometry (Siófok, 1985) Colloq. Math. Soc. János Bolyai, vol. 48, North-Holland, Amsterdam, 1987, pp. 551 – 571. · Zbl 0628.28006
[148] B. Uhrin, Curvilinear extensions of the Brunn-Minkowski-Lusternik inequality, Adv. Math. 109 (1994), no. 2, 288 – 312. · Zbl 0847.52007
[149] Richard A. Vitale, The Brunn-Minkowski inequality for random sets, J. Multivariate Anal. 33 (1990), no. 2, 286 – 293. · Zbl 0705.60013
[150] Richard A. Vitale, The translative expectation of a random set, J. Math. Anal. Appl. 160 (1991), no. 2, 556 – 562. · Zbl 0754.60021
[151] Roger Webster, Convexity, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994. · Zbl 0835.52001
[152] Gao Yong Zhang, Intersection bodies and the Busemann-Petty inequalities in \?\(^{4}\), Ann. of Math. (2) 140 (1994), no. 2, 331 – 346. · Zbl 0826.52011
[153] Gaoyong Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), no. 1, 183 – 202. · Zbl 1040.53089
[154] Gaoyong Zhang, A positive solution to the Busemann-Petty problem in \?\(^{4}\), Ann. of Math. (2) 149 (1999), no. 2, 535 – 543. · Zbl 0937.52004
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