The Brunn-Minkowski inequality

Gardner, R.

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

The Brunn-Minkowski inequality
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Bull. Amer. Math. Soc. 39 (2002), 355-405 Request permission

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.

References

S. Alesker, S. Dar, and V. Milman, A remarkable measure preserving diffeomorphism between two convex bodies in $\textbf {R}^n$, Geom. Dedicata 74 (1999), no. 2, 201–212. MR 1674116, DOI 10.1023/A:1005087216335
Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
Ben Andrews, Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138 (1999), no. 1, 151–161. MR 1714339, DOI 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 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 10.4153/CJM-1999-020-0

Atoms and Information Theory

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, DOI 10.1007/978-3-642-69881-1
Keith Ball, Logarithmically concave functions and sections of convex sets in $\textbf {R}^n$, Studia Math. 88 (1988), no. 1, 69–84. MR 932007, DOI 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 10.1007/BFb0090058
Keith Ball, Shadows of convex bodies, Trans. Amer. Math. Soc. 327 (1991), no. 2, 891–901. MR 1035998, DOI 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 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 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 10.1016/S0764-4442(97)86963-7

Inégalités Fonctionelles et Géométriques Obtenues par Transport des Mesures

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 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 10.1007/s000390050054
Franck Barthe, Restricted Prékopa-Leindler inequality, Pacific J. Math. 189 (1999), no. 2, 211–222. MR 1696120, DOI 10.2140/pjm.1999.189.211
Edwin F. Beckenbach and Richard Bellman, Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 30, Springer-Verlag, Inc., New York, 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 10.2307/1970980

Über die kleinste umbeschriebene und die grösste einbeschriebene Ellipse eines konvexen Bereichs

115

Determining convex bodies with piecewise ${C}^2$ boundary from their covariogram

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

From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities

10

Béla Bollobás and Imre Leader, Sums in the grid, Discrete Math. 162 (1996), no. 1-3, 31–48. MR 1425777, DOI 10.1016/S0012-365X(96)00303-2
Christer Borell, The Brunn-Minkowski inequality in Gauss space, Invent. Math. 30 (1975), no. 2, 207–216. MR 399402, DOI 10.1007/BF01425510
C. Borell, Convex set functions in $d$-space, Period. Math. Hungar. 6 (1975), no. 2, 111–136. MR 404559, DOI 10.1007/BF02018814
Christer Borell, Capacitary inequalities of the Brunn-Minkowski type, Math. Ann. 263 (1983), no. 2, 179–184. MR 698001, DOI 10.1007/BF01456879
Christer Borell, Geometric properties of some familiar diffusions in $\textbf {R}^n$, 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 10.1016/S0014-2921(98)00017-8
Christer Borell, Diffusion equations and geometric inequalities, Potential Anal. 12 (2000), no. 1, 49–71. MR 1745333, DOI 10.1023/A:1008641618547

Some inequalities for Gaussian measures and the long-range order of one-dimensional plasma

ed. by A. M. Arthurs

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 10.1016/0001-8708(76)90184-5
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 10.1016/0022-1236(76)90004-5
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, DOI 10.1007/978-94-017-1687-1
S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
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 10.1006/aima.1996.0008

Some applications of mass transport to Gaussian type inequalities

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 10.1016/S0764-4442(99)90008-3

A Riemannian interpolation inequality à la Borell, Brascamp and Lieb

146

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 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 10.1016/0022-247X(80)90136-5
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 10.1023/A:1005132006433
Somesh Das Gupta, Brunn-Minkowski inequality and its aftermath, J. Multivariate Anal. 10 (1980), no. 3, 296–318. MR 588074, DOI 10.1016/0047-259X(80)90051-2
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 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 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 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 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 10.4153/CJM-1961-037-0
Wm. J. Firey, $p$-means of convex bodies, Math. Scand. 10 (1962), 17–24. MR 141003, DOI 10.7146/math.scand.a-10510
William J. Firey, Shapes of worn stones, Mathematika 21 (1974), 1–11. MR 362045, DOI 10.1112/S0025579300005714
Irene Fonseca, The Wulff theorem revisited, Proc. Roy. Soc. London Ser. A 432 (1991), no. 1884, 125–145. MR 1116536, DOI 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 10.1017/S0308210500028365
B. Roy Frieden, Physics from Fisher information, Cambridge University Press, Cambridge, 1998. A unification. MR 1676801, DOI 10.1017/CBO9780511622670

The Brunn-Minkowski inequality: A survey with proofs

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 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 10.2307/2118606
Richard J. Gardner, Geometric tomography, Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, Cambridge, 1995. MR 1356221

A Brunn-Minkowski inequality for the integer lattice

353

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 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 10.2307/2373688

Information theoretic inequalities for contoured probability distributions

H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). MR 0102775
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1

Inequalities

Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
David Jerison, A Minkowski problem for electrostatic capacity, Acta Math. 176 (1996), no. 1, 1–47. MR 1395668, DOI 10.1007/BF02547334
Jeff Kahn and Nathan Linial, Balancing extensions via Brunn-Minkowski, Combinatorica 11 (1991), no. 4, 363–368. MR 1137768, DOI 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 10.1007/BF02574061

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
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
RafałLatała, A note on the Ehrhard inequality, Studia Math. 118 (1996), no. 2, 169–174. MR 1389763, DOI 10.4064/sm-118-2-169-174

Concentration of measure and logarithmic Sobolev inequalities

ed. by J. Azéma, M. Émery, M. Ledoux, and M. Yor, Lecture Notes in Mathematics 1709

The Concentration of Measure Phenomenon

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, DOI 10.1007/978-3-642-20212-4

On a certain converse of Hölder’s inequality. II

33

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 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, DOI 10.1090/gsm/014
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 10.1002/rsa.3240040402

Die Brunn-Minkowskische Ungleichung für beliebige messbare Mengen

8

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 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 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 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 10.1016/0001-8708(88)90077-1
Erwin Lutwak, Centroid bodies and dual mixed volumes, Proc. London Math. Soc. (3) 60 (1990), no. 2, 365–391. MR 1031458, DOI 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 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 10.1006/aima.1996.0022

The Brunn-Minkowski-Firey inequality for non-convex sets

The Cramer-Rao inequality for star bodies

112

On the ${L}_p$-Minkowski problem

Sharp affine ${L}_p$ Sobolev inequalities

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 10.1215/S0012-7094-00-10432-2

${L}_p$ affine isoperimetric inequalities

56

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, DOI 10.1017/CBO9780511623813
B. Maurey, Some deviation inequalities, Geom. Funct. Anal. 1 (1991), no. 2, 188–197. MR 1097258, DOI 10.1007/BF01896377

A Convexity Theory for Interacting Gases and Equilibrium Crystals

Robert J. McCann, A convexity principle for interacting gases, Adv. Math. 128 (1997), no. 1, 153–179. MR 1451422, DOI 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 10.1007/s002200050409
Peter McMullen, New combinations of convex sets, Geom. Dedicata 78 (1999), no. 1, 1–19. MR 1722167, DOI 10.1023/A:1005019802841

Inequalities in the sense of Brunn-Minkowski, Vitale for random convex bodies

Mathieu Meyer, Maximal hyperplane sections of convex bodies, Mathematika 46 (1999), no. 1, 131–136. MR 1750649, DOI 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 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, DOI 10.1007/978-1-4757-3845-2
Andrei Okounkov, Brunn-Minkowski inequality for multiplicities, Invent. Math. 125 (1996), no. 3, 405–411. MR 1400312, DOI 10.1007/s002220050081
Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182–1238. MR 500557, DOI 10.1090/S0002-9904-1978-14553-4

The geometry of dissipative evolution equations: the porous medium equation

26

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 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, DOI 10.1017/CBO9780511662454
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, DOI 10.1007/978-94-017-3087-7
Yosef Rinott, On convexity of measures, Ann. Probability 4 (1976), no. 6, 1020–1026. MR 428540, DOI 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 10.1023/A:1004958110076
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, DOI 10.1017/CBO9780511526282
J. Serra, Image analysis and mathematical morphology, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1984. English version revised by Noel Cressie. MR 753649
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336

Sur la question de la mesurabilité de la base de M. Hamel

1

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

The discrete planar $L_0$-Minkowski problem

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 10.1090/S0002-9904-1978-14499-1
Yung Liang Tong, Probability inequalities in multivariate distributions, Probabilities and Mathematical Statistics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. MR 572617
Neil S. Trudinger, Isoperimetric inequalities for quermassintegrals, Ann. Inst. H. Poincaré C Anal. Non Linéaire 11 (1994), no. 4, 411–425 (English, with English and French summaries). MR 1287239, DOI 10.1016/S0294-1449(16)30181-0
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 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 10.1016/0047-259X(90)90052-J
Richard A. Vitale, The translative expectation of a random set, J. Math. Anal. Appl. 160 (1991), no. 2, 556–562. MR 1126137, DOI 10.1016/0022-247X(91)90325-T
Roger Webster, Convexity, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994. MR 1443208
L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
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 10.2307/120974