The Brunn-Minkowski inequality

Gardner, R.

doi:10.1090/S0273-0979-02-00941-2

The Brunn-Minkowski inequality

Author: R. J. Gardner
Journal: Bull. Amer. Math. Soc. 39 (2002), 355-405
MSC (2000): Primary 26D15, 52A40
DOI: https://doi.org/10.1090/S0273-0979-02-00941-2
Published electronically: April 8, 2002
MathSciNet review: 1898210
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Abstract: In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of ${\mathbb R}^n$, and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications.

S. Alesker, S. Dar, and V. Milman, A remarkable measure preserving diffeomorphism between two convex bodies in ${\bf R}^n$, Geom. Dedicata 74 (1999), no. 2, 201–212. MR 1674116, DOI https://doi.org/10.1023/A%3A1005087216335

And55 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.

Ben Andrews, Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138 (1999), no. 1, 151–161. MR 1714339, DOI https://doi.org/10.1007/s002220050344
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. MR 1695717, DOI https://doi.org/10.1112/S0025579300014182
Hyoungsick Bahn and Paul Ehrlich, A Brunn-Minkowski type theorem on the Minkowski spacetime, Canad. J. Math. 51 (1999), no. 3, 449–469. MR 1701320, DOI https://doi.org/10.4153/CJM-1999-020-0

Bai71 R. Baierlein, Atoms and Information Theory, W. H. Freeman and Company, San Francisco, 1971.

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. MR 1305147
Keith Ball, Logarithmically concave functions and sections of convex sets in ${\bf R}^n$, Studia Math. 88 (1988), no. 1, 69–84. MR 932007, DOI https://doi.org/10.4064/sm-88-1-69-84
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. MR 1008726, DOI https://doi.org/10.1007/BFb0090058
Keith Ball, Shadows of convex bodies, Trans. Amer. Math. Soc. 327 (1991), no. 2, 891–901. MR 1035998, DOI https://doi.org/10.1090/S0002-9947-1991-1035998-3
Keith Ball, Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. (2) 44 (1991), no. 2, 351–359. MR 1136445, DOI https://doi.org/10.1112/jlms/s2-44.2.351
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. MR 1491097, DOI https://doi.org/10.2977/prims/1195164788
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). MR 1450443, DOI https://doi.org/10.1016/S0764-4442%2897%2986963-7

Bar97b ---, Inégalités Fonctionelles et Géométriques Obtenues par Transport des Mesures, Ph.D. thesis, Université de Marne-la-Vallée, Paris, 1997.

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). MR 1645730
Franck Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math. 134 (1998), no. 2, 335–361. MR 1650312, DOI https://doi.org/10.1007/s002220050267
F. Barthe, Optimal Young’s inequality and its converse: a simple proof, Geom. Funct. Anal. 8 (1998), no. 2, 234–242. MR 1616143, DOI https://doi.org/10.1007/s000390050054
Franck Barthe, Restricted Prékopa-Leindler inequality, Pacific J. Math. 189 (1999), no. 2, 211–222. MR 1696120, DOI https://doi.org/10.2140/pjm.1999.189.211
Edwin F. Beckenbach and Richard Bellman, Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., Bd. 30, Springer-Verlag, New York, Inc., 1965. Second revised printing. MR 0192009
William Beckner, Inequalities in Fourier analysis, Ann. of Math. (2) 102 (1975), no. 1, 159–182. MR 385456, DOI https://doi.org/10.2307/1970980

Beh38 F. Behrend, Über die kleinste umbeschriebene und die grösste einbeschriebene Ellipse eines konvexen Bereichs, Math. Ann. 115 (1938), 379–411.

Bia01 G. Bianchi, Determining convex bodies with piecewise ${C}^2$ boundary from their covariogram, preprint.

Nelson M. Blachman, The convolution inequality for entropy powers, IEEE Trans. Inform. Theory IT-11 (1965), 267–271. MR 188004, DOI https://doi.org/10.1109/tit.1965.1053768

BobL01 S. G. Bobkov and M. Ledoux, From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities, Geom. Funct. Anal. 10 (2000), 1028–1052.

Béla Bollobás and Imre Leader, Sums in the grid, Discrete Math. 162 (1996), no. 1-3, 31–48. MR 1425777, DOI https://doi.org/10.1016/S0012-365X%2896%2900303-2
Christer Borell, The Brunn-Minkowski inequality in Gauss space, Invent. Math. 30 (1975), no. 2, 207–216. MR 399402, DOI https://doi.org/10.1007/BF01425510
C. Borell, Convex set functions in $d$-space, Period. Math. Hungar. 6 (1975), no. 2, 111–136. MR 404559, DOI https://doi.org/10.1007/BF02018814
Christer Borell, Capacitary inequalities of the Brunn-Minkowski type, Math. Ann. 263 (1983), no. 2, 179–184. MR 698001, DOI https://doi.org/10.1007/BF01456879
Christer Borell, Geometric properties of some familiar diffusions in ${\bf R}^n$, Ann. Probab. 21 (1993), no. 1, 482–489. MR 1207234
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. MR 1665575, DOI https://doi.org/10.1016/S0014-2921%2898%2900017-8
Christer Borell, Diffusion equations and geometric inequalities, Potential Anal. 12 (2000), no. 1, 49–71. MR 1745333, DOI https://doi.org/10.1023/A%3A1008641618547

BraL75 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.

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. MR 412366, DOI https://doi.org/10.1016/0001-8708%2876%2990184-5
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. MR 0450480, DOI https://doi.org/10.1016/0022-1236%2876%2990004-5
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. MR 1784349

BurZ88 Y. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer, New York, 1988. Russian original: 1980. ;

Bus49 H. Busemann, The isoperimetric problem for Minkowski area, Amer. J. Math. 71 (1949), 743–762.

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. MR 1371649, DOI https://doi.org/10.1006/aima.1996.0008

Cor01 D. Cordero-Erausquin, Some applications of mass transport to Gaussian type inequalities, Arch. Rational Mech. Anal., to appear.

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). MR 1724541, DOI https://doi.org/10.1016/S0764-4442%2899%2990008-3

CMS00 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.

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. MR 782217, DOI https://doi.org/10.1109/TIT.1984.1056983
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. MR 572660, DOI https://doi.org/10.1016/0022-247X%2880%2990136-5
I. Dancs and B. Uhrin, On the conditions of equality in an integral inequality, Publ. Math. Debrecen 29 (1982), no. 1-2, 117–132. MR 673145
S. Dar, A Brunn-Minkowski-type inequality, Geom. Dedicata 77 (1999), no. 1, 1–9. MR 1706512, DOI https://doi.org/10.1023/A%3A1005132006433
Somesh Das Gupta, Brunn-Minkowski inequality and its aftermath, J. Multivariate Anal. 10 (1980), no. 3, 296–318. MR 588074, DOI https://doi.org/10.1016/0047-259X%2880%2990051-2
Amir Dembo, Thomas M. Cover, and Joy A. Thomas, Information-theoretic inequalities, IEEE Trans. Inform. Theory 37 (1991), no. 6, 1501–1518. MR 1134291, DOI https://doi.org/10.1109/18.104312
Sudhakar Dharmadhikari and Kumar Joag-Dev, Unimodality, convexity, and applications, Probability and Mathematical Statistics, Academic Press, Inc., Boston, MA, 1988. MR 954608
Alexander Dinghas, Über eine Klasse superadditiver Mengenfunktionale von Brunn-Minkowski-Lusternikschem Typus, Math. Z. 68 (1957), 111–125 (German). MR 96173, DOI https://doi.org/10.1007/BF01160335
Richard M. Dudley, Real analysis and probability, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1989. MR 982264
R. M. Dudley, Metric marginal problems for set-valued or non-measurable variables, Probab. Theory Related Fields 100 (1994), no. 2, 175–189. MR 1296427, DOI https://doi.org/10.1007/BF01199264
H. G. Eggleston, Convexity, Cambridge Tracts in Mathematics and Mathematical Physics, No. 47, Cambridge University Press, New York, 1958. MR 0124813
Antoine Ehrhard, Symétrisation dans l’espace de Gauss, Math. Scand. 53 (1983), no. 2, 281–301 (French). MR 745081, DOI https://doi.org/10.7146/math.scand.a-12035
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). MR 850753
Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
William J. Firey, Polar means of convex bodies and a dual to the Brunn-Minkowski theorem, Canadian J. Math. 13 (1961), 444–453. MR 177350, DOI https://doi.org/10.4153/CJM-1961-037-0
Wm. J. Firey, $p$-means of convex bodies, Math. Scand. 10 (1962), 17–24. MR 141003, DOI https://doi.org/10.7146/math.scand.a-10510
William J. Firey, Shapes of worn stones, Mathematika 21 (1974), 1–11. MR 362045, DOI https://doi.org/10.1112/S0025579300005714
Irene Fonseca, The Wulff theorem revisited, Proc. Roy. Soc. London Ser. A 432 (1991), no. 1884, 125–145. MR 1116536, DOI https://doi.org/10.1098/rspa.1991.0009
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. MR 1130601, DOI https://doi.org/10.1017/S0308210500028365
B. Roy Frieden, Physics from Fisher information, Cambridge University Press, Cambridge, 1998. A unification. MR 1676801

Gar01 R. J. Gardner, The Brunn-Minkowski inequality: A survey with proofs, available at http://www.ac.wwu.edu/~gardner.

R. J. Gardner, Intersection bodies and the Busemann-Petty problem, Trans. Amer. Math. Soc. 342 (1994), no. 1, 435–445. MR 1201126, DOI https://doi.org/10.1090/S0002-9947-1994-1201126-7
R. J. Gardner, A positive answer to the Busemann-Petty problem in three dimensions, Ann. of Math. (2) 140 (1994), no. 2, 435–447. MR 1298719, DOI https://doi.org/10.2307/2118606
Richard J. Gardner, Geometric tomography, Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, Cambridge, 1995. MR 1356221

GarG99 R. J. Gardner and P. Gronchi, A Brunn-Minkowski inequality for the integer lattice, Trans. Amer. Math. Soc. 353 (2001), 3995–4024.

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. MR 1689343, DOI https://doi.org/10.2307/120978
R. J. Gardner and Gaoyong Zhang, Affine inequalities and radial mean bodies, Amer. J. Math. 120 (1998), no. 3, 505–528. MR 1623396
H. Groemer, Stability of geometric inequalities, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 125–150. MR 1242978
M. Gromov, Convex sets and Kähler manifolds, Advances in differential geometry and topology, World Sci. Publ., Teaneck, NJ, 1990, pp. 1–38. MR 1095529
Leonard Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061–1083. MR 420249, DOI https://doi.org/10.2307/2373688

GLYZ00 O.-G. Guleryuz, E. Lutwak, D. Yang, and G. Zhang, Information theoretic inequalities for contoured probability distributions, preprint.

H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). MR 0102775

HadO56 H. Hadwiger and D. Ohmann, Brunn-Minkowskischer Satz und Isoperimetrie, Math. Zeit. 66 (1956), 1–8.

HLP59 G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1959.

HenM53 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.

David Jerison, A Minkowski problem for electrostatic capacity, Acta Math. 176 (1996), no. 1, 1–47. MR 1395668, DOI https://doi.org/10.1007/BF02547334
Jeff Kahn and Nathan Linial, Balancing extensions via Brunn-Minkowski, Combinatorica 11 (1991), no. 4, 363–368. MR 1137768, DOI https://doi.org/10.1007/BF01275670
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. MR 1318794, DOI https://doi.org/10.1007/BF02574061

King S. P. King, website at http://members.home.net/stephenk1/Outlaw/fisherinfo.html.

J. F. C. Kingman and S. J. Taylor, Introduction to measure and probability, Cambridge University Press, London-New York-Ibadan, 1966. MR 0218515

Kno57 H. Knothe, Contributions to the theory of convex bodies, Michigan Math. J. 4 (1957), 39–52.

Rafał Latała, A note on the Ehrhard inequality, Studia Math. 118 (1996), no. 2, 169–174. MR 1389763, DOI https://doi.org/10.4064/sm-118-2-169-174

Led99 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.

Led01 ---, The Concentration of Measure Phenomenon, American Mathematical Society, Providence, RI, 2001.

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. MR 1102015

Lei72 L. Leindler, On a certain converse of Hölder’s inequality. II, Acta Sci. Math. (Szeged) 33 (1972), 217–223.

Elliott H. Lieb, Proof of an entropy conjecture of Wehrl, Comm. Math. Phys. 62 (1978), no. 1, 35–41. MR 506364
Elliott H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990), no. 1, 179–208. MR 1069246, DOI https://doi.org/10.1007/BF01233426
Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225
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. MR 1243006
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. MR 1238906, DOI https://doi.org/10.1002/rsa.3240040402

Lus35 L. A. Lusternik, Die Brunn-Minkowskische Ungleichung für beliebige messbare Mengen, C. R. (Doklady) Acad. Sci. URSS 8 (1935), 55–58.

Erwin Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), no. 2, 531–538. MR 380631
Erwin Lutwak, Width-integrals of convex bodies, Proc. Amer. Math. Soc. 53 (1975), no. 2, 435–439. MR 383254, DOI https://doi.org/10.1090/S0002-9939-1975-0383254-5
Erwin Lutwak, A general isepiphanic inequality, Proc. Amer. Math. Soc. 90 (1984), no. 3, 415–421. MR 728360, DOI https://doi.org/10.1090/S0002-9939-1984-0728360-3
Erwin Lutwak, Volume of mixed bodies, Trans. Amer. Math. Soc. 294 (1986), no. 2, 487–500. MR 825717, DOI https://doi.org/10.1090/S0002-9947-1986-0825717-3
Erwin Lutwak, Intersection bodies and dual mixed volumes, Adv. in Math. 71 (1988), no. 2, 232–261. MR 963487, DOI https://doi.org/10.1016/0001-8708%2888%2990077-1
Erwin Lutwak, Centroid bodies and dual mixed volumes, Proc. London Math. Soc. (3) 60 (1990), no. 2, 365–391. MR 1031458, DOI https://doi.org/10.1112/plms/s3-60.2.365
Erwin Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), no. 1, 131–150. MR 1231704
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). MR 1191884
Erwin Lutwak, Selected affine isoperimetric inequalities, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 151–176. MR 1242979, DOI https://doi.org/10.1016/B978-0-444-89596-7.50010-9
Erwin Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math. 118 (1996), no. 2, 244–294. MR 1378681, DOI https://doi.org/10.1006/aima.1996.0022

LYZ01 E. Lutwak, D. Yang, and G. Zhang, The Brunn-Minkowski-Firey inequality for non-convex sets, preprint.

LYZ00c ---, The Cramer-Rao inequality for star bodies, Duke Math. J. 112 (2002), 59–81.

LYZ00b ---, On the ${L}_p$-Minkowski problem, preprint.

LYZ01b ---, Sharp affine ${L}_p$ Sobolev inequalities, preprint.

Erwin Lutwak, Deane Yang, and Gaoyong Zhang, A new ellipsoid associated with convex bodies, Duke Math. J. 104 (2000), no. 3, 375–390. MR 1781476, DOI https://doi.org/10.1215/S0012-7094-00-10432-2

LYZ001 ---, ${L}_p$ affine isoperimetric inequalities, J. Diff. Geom. 56 (2000), 111–132.

Erwin Lutwak and Gaoyong Zhang, Blaschke-Santaló inequalities, J. Differential Geom. 47 (1997), no. 1, 1–16. MR 1601426
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. MR 1333890
B. Maurey, Some deviation inequalities, Geom. Funct. Anal. 1 (1991), no. 2, 188–197. MR 1097258, DOI https://doi.org/10.1007/BF01896377

McC94 R. J. McCann, A Convexity Theory for Interacting Gases and Equilibrium Crystals, Ph.D. dissertation, Princeton University, 1994.

Robert J. McCann, A convexity principle for interacting gases, Adv. Math. 128 (1997), no. 1, 153–179. MR 1451422, DOI https://doi.org/10.1006/aima.1997.1634
Robert J. McCann, Equilibrium shapes for planar crystals in an external field, Comm. Math. Phys. 195 (1998), no. 3, 699–723. MR 1641031, DOI https://doi.org/10.1007/s002200050409
Peter McMullen, New combinations of convex sets, Geom. Dedicata 78 (1999), no. 1, 1–19. MR 1722167, DOI https://doi.org/10.1023/A%3A1005019802841

MecS01 J. Mecke and A. Schwella, Inequalities in the sense of Brunn-Minkowski, Vitale for random convex bodies, preprint.

Mathieu Meyer, Maximal hyperplane sections of convex bodies, Mathematika 46 (1999), no. 1, 131–136. MR 1750649, DOI https://doi.org/10.1112/S0025579300007622
V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed $n$-dimensional space, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 64–104. MR 1008717, DOI https://doi.org/10.1007/BFb0090049
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. MR 856576
Melvyn B. Nathanson, Additive number theory, Graduate Texts in Mathematics, vol. 165, Springer-Verlag, New York, 1996. Inverse problems and the geometry of sumsets. MR 1477155
Andrei Okounkov, Brunn-Minkowski inequality for multiplicities, Invent. Math. 125 (1996), no. 3, 405–411. MR 1400312, DOI https://doi.org/10.1007/s002220050081
Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182–1238. MR 500557, DOI https://doi.org/10.1090/S0002-9904-1978-14553-4

Ott01 F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations 26 (2001), 101–174.

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. MR 1760620, DOI https://doi.org/10.1006/jfan.1999.3557
Gilles Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989. MR 1036275
András Prékopa, Logarithmic concave measures with application to stochastic programming, Acta Sci. Math. (Szeged) 32 (1971), 301–316. MR 315079
András Prékopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged) 34 (1973), 335–343. MR 404557
András Prékopa, Stochastic programming, Mathematics and its Applications, vol. 324, Kluwer Academic Publishers Group, Dordrecht, 1995. MR 1375234
Yosef Rinott, On convexity of measures, Ann. Probability 4 (1976), no. 6, 1020–1026. MR 428540, DOI https://doi.org/10.1214/aop/1176995947
Imre Z. Ruzsa, The Brunn-Minkowski inequality and nonconvex sets, Geom. Dedicata 67 (1997), no. 3, 337–348. MR 1475877, DOI https://doi.org/10.1023/A%3A1004958110076
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. MR 1196706
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. MR 1665592
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521
J. Serra, Image analysis and mathematical morphology, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1984. English version revised by Noel Cressie. MR 753649

Sha48 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.

Sie20 W. Sierpiński, Sur la question de la mesurabilité de la base de M. Hamel, Fund. Math. 1 (1920), 105–111.

A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Information and Control 2 (1959), 101–112. MR 109101

Sta01 A. Stancu, The discrete planar $L_0$-Minkowski problem, Adv. Math., to appear.

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. MR 895588
V. N. Sudakov and B. S. Cirel′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. MR 0365680
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. MR 1395205
Jean E. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc. 84 (1978), no. 4, 568–588. MR 493671, DOI https://doi.org/10.1090/S0002-9904-1978-14499-1
Yung Liang Tong, Probability inequalities in multivariate distributions, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. Probabilities and Mathematical Statistics. MR 572617
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). MR 1287239, DOI https://doi.org/10.1016/S0294-1449%2816%2930181-0
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. MR 910735
B. Uhrin, Curvilinear extensions of the Brunn-Minkowski-Lusternik inequality, Adv. Math. 109 (1994), no. 2, 288–312. MR 1304754, DOI https://doi.org/10.1006/aima.1994.1088
Richard A. Vitale, The Brunn-Minkowski inequality for random sets, J. Multivariate Anal. 33 (1990), no. 2, 286–293. MR 1055274, DOI https://doi.org/10.1016/0047-259X%2890%2990052-J
Richard A. Vitale, The translative expectation of a random set, J. Math. Anal. Appl. 160 (1991), no. 2, 556–562. MR 1126137, DOI https://doi.org/10.1016/0022-247X%2891%2990325-T
Roger Webster, Convexity, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994. MR 1443208

Zha94 Gaoyong Zhang, Intersection bodies and the Busemann-Petty inequalities in ${\mathbb {R}}^4$, Ann. of Math. 140 (1994), 331–346.

Gaoyong Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), no. 1, 183–202. MR 1776095
Gaoyong Zhang, A positive solution to the Busemann-Petty problem in $\mathbf R^4$, Ann. of Math. (2) 149 (1999), no. 2, 535–543. MR 1689339, DOI https://doi.org/10.2307/120974