## The Brunn-Minkowski inequality (2002)

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Venue: | Bull. Amer. Math. Soc. (N.S |

Citations: | 90 - 5 self |

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@ARTICLE{Gardner02thebrunn-minkowski,

author = {R. J. Gardner},

title = {The Brunn-Minkowski inequality},

journal = {Bull. Amer. Math. Soc. (N.S},

year = {2002},

volume = {39},

pages = {355--405}

}

### Years of Citing Articles

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### Abstract

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 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. 1.

### Citations

6983 |
The Mathematical Theory of Communication
- SHANNON, WEAVER
- 1949
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Citation Context ...te to physics and logarithmic Sobolev inequalities. Suppose that X is a discrete random variable taking possible values x1,...,xm with probabilities p1,...,pm, respectively, where � i pi = 1. Shannon =-=[137]-=- introduced a measure of the average uncertainty removed by revealing the value of X. This quantity, m� Hm(p1,...,pm) =− pi log pi, is called the entropy of X. It can also be regarded as a measure of ... |

1217 | Image Analysis and Mathematical Morphology - Serra - 1982 |

901 | Geometric Measure Theory - Federer - 1969 |

755 | Measure theory and Fine properties of functions - Evans, Gariepy - 1992 |

668 | Convex Bodies. The Brunn–Minkowski Theory - Schneider - 1993 |

483 | Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability - Mattila - 1995 |

310 | The Volume of Convex Bodies and Banach Space Geometry - Pisier - 1989 |

299 | Probability in Banach Spaces - Ledoux, Talagrand - 1991 |

260 |
Logarithmic Sobolev inequalities
- Gross
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Citation Context ... and Thomas [47] explore several related inequalities involving entropy, Fisher information, and uncertainty principles. Another rich area surrounds the logarithmic Sobolev inequality proved by Gross =-=[72]-=-: Entγn(f) ≤ 1 2 Iγn(f), (58) where f is a suitably smooth nonnegative function on Rn , γn is the Gauss measure in Rn defined by (44), � Entγn(f) = Rn �� f(x)logf(x) dγn(x) − Rn ��� f(x) dγn(x) Rn � l... |

254 | The geometry of dissipative evolution equations: the porous medium equation
- Otto
- 2001
(Show Context)
Citation Context ...x) that depend also on a parameter σ. This work yields a “Brownian motion” proof of the Brunn-Minkowski inequality. McCann’s displacement convexity (30) plays an essential role in recent work of Otto =-=[125]-=-, who observed that various diffusion equations can be viewed as gradient flows in the space of probability measures with the Wasserstein metric (formally, at least, an infinite-dimensional Riemannian... |

187 | Asymptotic Theory of Finite Dimensional Normed Spaces - Milman, Schechtman - 1986 |

152 |
Inequalities in Fourier analysis
- Beckner
- 1975
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Citation Context ...es and Hölder’s inequality (25) suffice, as in [91, p. 99]. The next theorem provides two convolution inequalities with sharp constants, the first a sharp form of (48) proved independently by Beckner =-=[20]-=- and Brascamp and Lieb [33], and the second a reverse form found by Brascamp and Lieb [33] (refining an earlier version due to Leindler [88]). Theorem 13.1. Let 0 < p,q,r satisfy (47), andletf ∈ L p (... |

150 |
Vorlesungen uber Inhalt, Oberflache und Isoperimetrie
- Hadwiger
- 1957
(Show Context)
Citation Context ...0 <λ<1andsetsX, Y in the class. For example, the Brunn-Minkowski inequality (2) implies that V 1/n n is Minkowski concave on the class of convex bodies. When Hadwiger published his extraordinary book =-=[74]-=- in 1957, many other Minkowski-concave functions had already been found, and several more have been discovered since. We shall present some of these; all the functions have the required degree of posi... |

139 |
Additive Number Theory. Inverse Problems and the Geometry of
- Nathanson
- 1996
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Citation Context ...mpty finite subsets of Z/pZ, then |X + Y |≥min{p, |X| + |Y |−1}. Here |X| is the cardinality of X. Many generalizations of this result, including Kneser’s extension to Abelian groups, are surveyed in =-=[122]-=-. The lower bound for a vector sum is in the spirit of the Brunn-Minkowski inequality. We now describe a closer analog. Let Y be a finite subset of Z n with |Y |≥n +1. Forx =(x1,...,xn) ∈ Z n ,let wY ... |

130 | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality
- Otto, Villani
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Citation Context ...of Brascamp and Lieb proved in [34]). Cordero-Erausquin [39] proves (58) directly using the transport of mass idea from Section 8. McCann’s displacement convexity (30) is utilized by Otto and Villani =-=[126]-=-, who find a new proof of an inequality of Talagrand for the Wasserstein distance between two probability measures in an n-dimensional Riemannian manifold, and show that Talagrand’s inequality is very... |

123 | Random walks in a convex body and an improved volume algorithm. Random Structures and Algorithms - Lovász, Simonovits - 1993 |

123 | Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space - Milman, Pajor - 1989 |

113 | Geometric Inequalities - Burago - 1988 |

103 |
Concentration of measure and logarithmic Sobolev inequalities. Séminaire de
- Ledoux
- 1999
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Citation Context ...of measure phenomenon that Milman applied in his 1971 proof of Dvoretzky’s theorem and that with contributions by Talagrand and others has quickly generated an extensive literature surveyed by Ledoux =-=[85]-=-, [86]. An excellent, but more selective, introduction is Ball’s elegant and insightful expository article [12, Lecture 8]. Analogous results hold in Gauss space, R n with the usual metric but with th... |

100 |
The Brunn-Minkowski inequality in Gauss Spaces
- Borell
- 1974
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Citation Context ... and an abundance of additional information and references can be found in Ledoux and Talagrand’s book [87, Section 1.1]. To describe some of this work briefly, let Φ(r) =γ1((−∞,r)) for r ∈ R. Borell =-=[26]-=- and Sudakov and Tsirel’son [142] independently showed that if X is a measurable subset of R n and γn(X) =Φ(rX), then γn(Xε) ≥ Φ(rX + ε), with equality if X is a half-space. Ehrhard [53], [54] gave a ... |

90 |
Information theoretic inequalities
- Dembo, Cover, et al.
- 1991
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Citation Context ...ws that equality holds in (55) if X and Y are multivariate normal with proportional covariances. In fact equality holds only for such X and Y , as Stam’s different proof [139] (simplified in [23] and =-=[47]-=-) of (55) shows. The most accessible direct proof of (55) seems to be that of Blachman [23]. As Lieb [89] discovered, however, the limiting case r → 1 of Young’s inequality (49) yields the entropy pow... |

85 |
The isoperimetric inequality
- Osserman
- 1978
(Show Context)
Citation Context ...THEMATICAL SOCIETY Volume 39, Number 3, Pages 355–405 S 0273-0979(02)00941-2 Article electronically published on April 8, 2002 THE BRUNN-MINKOWSKI INEQUALITY R. J. GARDNER 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 s... |

84 | Logarithmic concave measures with applications to stochastic programming
- Prékopa
- 1971
(Show Context)
Citation Context ... G λ . The proof for general n is just as accessible. This is by induction on n and can be found in [63, Theorem 4.2]. The Prékopa-Leindler inequality (21) was explicitly stated and proved by Prékopa =-=[128]-=-, [129] and Leindler [88]. (See the historical remarks after Theorem 10.1, however.) There are two basic ingredients in the above proof: the introduction in (28) of the volume parameter t, and use of ... |

79 |
The convolution inequality for entropy powers
- Blachman
- 1965
(Show Context)
Citation Context ...proof shows that equality holds in (55) if X and Y are multivariate normal with proportional covariances. In fact equality holds only for such X and Y , as Stam’s different proof [139] (simplified in =-=[23]-=- and [47]) of (55) shows. The most accessible direct proof of (55) seems to be that of Blachman [23]. As Lieb [89] discovered, however, the limiting case r → 1 of Young’s inequality (49) yields the en... |

76 | Isoperimetric problems for convex bodies and a localization lemma, Discrete Comput
- Kannan, Lovász, et al.
- 1995
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Citation Context ...lity is very closely related to one found earlier by Ball [8]. For other associated inequalities, see [69, Theorem 4.1] and [119, Lemma 1]. 18.11. Further applications. Kannan, Lovász, and Simonovits =-=[80]-=- obtain some inequalities involving log-concave functions by means of a “localization lemma” that reduces certain inequalities involving integrals over convex bodies in R n to integral inequalities ov... |

59 |
Probability Inequalities in Multivariate Distributions
- Tong
- 1980
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Citation Context ... J. GARDNER Anderson’s theorem has many applications in probability and statistics, where, for example, it can be applied to show that certain statistical tests are unbiased. See [2], [35], [48], and =-=[145]-=-. Certain of these applications are also associated with the Prékopa-Leindler inequality (21) and its generalization, the Borell-Brascamp-Lieb inequality (38). In Section 9 it was shown that the Préko... |

58 | A Riemannian interpolation inequality à la Borell, Brascamp and
- Cordero-Erausquin, McCann, et al.
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Citation Context ...as in the case p = 0 (see (36)). Dancs and Uhrin [43] also offer a version of Theorem 10.1 for −∞ ≤ p<−1/n. In calling Theorem 10.1 the Borell-Brascamp-Lieb inequality we are following the authors of =-=[41]-=- (who also generalize it to a Riemannian manifold setting; see Section 12) and placing the emphasis on the negative values of p. Infact,itcanbe shown (see [41] and [63, Section 10]) that Theorem 10.1 ... |

55 |
The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities
- Anderson
- 1955
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Citation Context ...inequality has as ingredients two kinds of curvilinear convex combinations of vectors, and its proof reintroduces geometrical methods. 11. Applications to probability and statistics In 1955, Anderson =-=[2]-=- used the Brunn-Minkowski inequality in his work on multivariate unimodality. He began with the following simple observation. If a nonnegative integrable function f on R is (i) symmetric (f(x) =f(−x))... |

54 |
Logarithmically concave functions and sections of convex sets
- Ball
- 1988
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Citation Context ...tive x, y ∈ R. Then � ∞ 0 h(x) dx ≥ M−p �� ∞ 0 (1−λ)yp (1−λ)yp +λxp λx g(y) p (1−λ)yp +λxp , f(x) dx, � ∞ 0 � g(x) dx, λ . The previous inequality is very closely related to one found earlier by Ball =-=[8]-=-. For other associated inequalities, see [69, Theorem 4.1] and [119, Lemma 1]. 18.11. Further applications. Kannan, Lovász, and Simonovits [80] obtain some inequalities involving log-concave functions... |

49 |
Unimodality, Convexity, and Applications
- Dharmadhikari, Joag-Dev
- 1988
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Citation Context ...4 .s378 R. J. GARDNER Anderson’s theorem has many applications in probability and statistics, where, for example, it can be applied to show that certain statistical tests are unbiased. See [2], [35], =-=[48]-=-, and [145]. Certain of these applications are also associated with the Prékopa-Leindler inequality (21) and its generalization, the Borell-Brascamp-Lieb inequality (38). In Section 9 it was shown tha... |

46 |
Crystalline variational problem
- Taylor
- 1978
(Show Context)
Citation Context ... continuous, and Fonseca [60] and Fonseca and Müller [61] extended the results to include sets M of finite perimeter in Rn . Good introductions with more details and references are provided by Taylor =-=[144]-=- and McCann [116]. In fact, McCann [116] also proves more general results that incorporate a convex external potential, by a technique developed in his paper [115] on interacting gases; see Section 8.... |

44 | Introduction to Measure and Probability - Kingman, Taylor - 1966 |

43 | From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities, Geometric and Functional Analysis 10
- Bobkov, Ledoux
- 2000
(Show Context)
Citation Context ...equalities is provided by Lieb and Loss [91, Chapter 8], where it is shown that they can be deduced from Young’s inequality (49) and used to estimate solutions of the heat equation. Bobkov and Ledoux =-=[24]-=- derive (58) from the Prékopa-Leindler inequality (the “Brascamp-Lieb” in the title of [24] refers not to (59) below but to a different inequality of Brascamp and Lieb proved in [34]). Cordero-Erausqu... |

42 |
Physics from Fisher Information: A unification
- Frieden
- 1998
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Citation Context ...vex geometry. An important concept called Fisher information was employed by Stam [139] in his proof of (55). Named after the statistician R. A. Fisher, Fisher information is claimed in a recent book =-=[62]-=- by Frieden to be at the heart of a unifying principle for all of physics! If X is a random variable with probability density f on R, the Fisher information I(X) ofX is � I(X) =I(f) =− R f(x)(log f(x)... |

42 | An analytic solution to the BusemannPetty problem on sections of convex bodies
- Gardner, Koldobsky, et al.
- 1999
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Citation Context ...n is always smaller in volume than that of another such body, is its volume also smaller? The answer is no in general in five or more dimensions, but yes in less than five dimensions. See [64], [65], =-=[68]-=-, [152], and [154]. Lutwak [95] also discovered that integrals over Sn−1 of products of radial functions behave like mixed volumes and called them dual mixed volumes. In the same paper, he showed that... |

41 |
On a certain converse of Hölder’s inequality
- Leindler
- 1972
(Show Context)
Citation Context ...al n is just as accessible. This is by induction on n and can be found in [63, Theorem 4.2]. The Prékopa-Leindler inequality (21) was explicitly stated and proved by Prékopa [128], [129] and Leindler =-=[88]-=-. (See the historical remarks after Theorem 10.1, however.) There are two basic ingredients in the above proof: the introduction in (28) of the volume parameter t, and use of the arithmetic-geometric ... |

40 | Analysis (second edition - Lieb, Loss - 2001 |

40 |
Intersection bodies and the Busemann-Petty problem
- Gardner
(Show Context)
Citation Context ...ng the origin is always smaller in volume than that of another such body, is its volume also smaller? The answer is no in general in five or more dimensions, but yes in less than five dimensions. See =-=[65]-=-, [66], [69], [151], and [153]. Lutwak [95] also discovered that integrals over S n\Gamma1 of products of radial functions behave like mixed volumes, and called them dual mixed volumes. In the same pa... |

39 |
Extremal properties of half{spaces for spherically invariant measures
- Sudakov, Tsirel'son
- 1974
(Show Context)
Citation Context ...information and references can be found in Ledoux and Talagrand’s book [87, Section 1.1]. To describe some of this work briefly, let Φ(r) =γ1((−∞,r)) for r ∈ R. Borell [26] and Sudakov and Tsirel’son =-=[142]-=- independently showed that if X is a measurable subset of R n and γn(X) =Φ(rX), then γn(Xε) ≥ Φ(rX + ε), with equality if X is a half-space. Ehrhard [53], [54] gave a new proof using symmetrization te... |

33 |
Dual mixed volumes
- Lutwak
- 1975
(Show Context)
Citation Context ...han that of another such body, is its volume also smaller? The answer is no in general in five or more dimensions, but yes in less than five dimensions. See [64], [65], [68], [152], and [154]. Lutwak =-=[95]-=- also discovered that integrals over Sn−1 of products of radial functions behave like mixed volumes and called them dual mixed volumes. In the same paper, he showed that a suitable version of Hölder’s... |

29 | Convex analysis and nonlinear geometric elliptic equations - Bakelman - 1994 |

29 |
Some deviation inequalities
- Maurey
- 1991
(Show Context)
Citation Context ...ger than (45) for convex bodies, it is unknown whether it holds for Borel sets; see [84] and [87, Problem 1]. An approximate isoperimetric inequality similar to (43) also holds in Gauss space; Maurey =-=[113-=-] (see also see [12, Theorem 8.1]) found a simple proof employing the Prekopa-Leindler inequality (21). As in S n 1 , there is a concentration of measure in Gauss space, this time in spherical shells ... |

28 | Real Analysis and Probability Wadsworth & Brooks/Cole - Dudley - 1989 |

26 |
Intersection bodies and the Busemann-Petty inequalities in R4
- Zhang
- 1994
(Show Context)
Citation Context ...lways smaller in volume than that of another such body, is its volume also smaller? The answer is no in general in five or more dimensions, but yes in less than five dimensions. See [64], [65], [68], =-=[152]-=-, and [154]. Lutwak [95] also discovered that integrals over Sn−1 of products of radial functions behave like mixed volumes and called them dual mixed volumes. In the same paper, he showed that a suit... |

25 |
The Wulff theorem revisited
- Fonseca
- 1991
(Show Context)
Citation Context ... done first by A. Dinghas in 1943 for convex polygons and polyhedra and then by various people in greater generality. In particular, Busemann [37] solved the problem when f is continuous, and Fonseca =-=[60]-=- and Fonseca and Müller [61] extended the results to include sets M of finite perimeter in Rn . Good introductions with more details and references are provided by Taylor [144] and McCann [116]. In fa... |

24 |
Contributions to the theory of convex bodies
- Knothe
- 1957
(Show Context)
Citation Context ...the required degree of positive homogeneity to allow the coefficients (1 − λ) andλto be deleted in (71). Other examples can be found in [74, Section 6.4] and in Lutwak’s papers [96] and [102]. Knothe =-=[83]-=- gave a proof of the Brunn-Minkowski inequality (2) for convex bodies, sketched in [135, pp. 312–314], and the following generalization. For each convex body K in Rn ,letF (K, x), x ∈ K, be a nonnegat... |

23 |
Some applications of mass transport to Gaussian type inequalities
- Cordero-Erausquin
- 2002
(Show Context)
Citation Context ...rive (58) from the Prékopa-Leindler inequality (the “Brascamp-Lieb” in the title of [24] refers not to (59) below but to a different inequality of Brascamp and Lieb proved in [34]). Cordero-Erausquin =-=[39]-=- proves (58) directly using the transport of mass idea from Section 8. McCann’s displacement convexity (30) is utilized by Otto and Villani [126], who find a new proof of an inequality of Talagrand fo... |

23 |
Über eine Klasse superadditiver Mengenfunktionale von Brunn-Minkowski-Lusternikschem Typus
- Dinghas
- 1957
(Show Context)
Citation Context ...in Section 7 for the proof of Theorem 7.1; see [63, Section 10] for the details. The result was first proved (in slightly modified form) for p>0 by Henstock and Macbeath [77] (when n = 1) and Dinghas =-=[49]-=-. The limiting case p =0wasalsoproved by Prékopa and Leindler, as noted above, and rediscovered by Brascamp and Lieb [32]. In general form Theorem 10.1 is stated and proved by Brascamp and Lieb [34, T... |

22 |
Proof of an entropy conjecture of Wehrl
- Lieb
- 1978
(Show Context)
Citation Context ...equality holds only for such X and Y , as Stam’s different proof [139] (simplified in [23] and [47]) of (55) shows. The most accessible direct proof of (55) seems to be that of Blachman [23]. As Lieb =-=[89]-=- discovered, however, the limiting case r → 1 of Young’s inequality (49) yields the entropy power inequality (55). A complete proof of this arresting fact can be found in [89] (or see [63, Section 18]... |

21 |
S.Müller, A uniqueness proof for the Wulff Theorem
- Fonseca
- 1991
(Show Context)
Citation Context ...pproximation arguments. In fact, Fonseca’s result is more general (see the next section on Wulff shape of crystals). A strong form of the Brunn-Minkowski inequality is also used by Fonseca and Müller =-=[61]-=-, again in the more general context of Wulff shape, to establish the corresponding equality conditions (the same as for (7)). The distinction between geometry and analysis is blurred even at the level... |