 (Parte 1 de 10)

THE BRUNN-MINKOWSKI INEQUALITY OCTOBER 25, 2001

Abstract. In 1978, Osserman  wrote a rather comprehensive 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 Rn, and deserves to be better known. We present a guide that explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some of its recent applications.

1. Introduction

About a century ago, not long after the first complete proof of the classical isoperimetric inequality was found, Minkowski proved the following inequality:

Here K and L are convex bodies (compact convex sets with nonempty interiors) in Rn, 0 < λ < 1, V denotes volume, and + denotes vector or Minkowski sum. The inequality (1) had been proved for n = 3 earlier by Brunn, and now it is known as the Brunn-Minkowski inequality. It is a sharp inequality, equality holding if and only if K and L are homothetic.

The Brunn-Minkowski inequality was inspired by issues around the isoperimetric problem, and was for a long time considered to belong to geometry, where its significance is widely recognized. It implies, but is much stronger than, the intuitively clear fact that the function that gives the volumes of parallel hyperplane sections of a convex body is unimodal. It can be proved on a single page (see Section 6), yet it quickly yields the classical isoperimetric inequality (21) for convex bodies and other important classes of sets. The fundamental geometric content of the Brunn-Minkowski inequality 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, surface area, and mean width.

By the mid-twentieth century, however, when Lusternik, Hadwiger and Ohmann, and Henstock and Macbeath had established a satisfactory generalization of (1) and its equality conditions to Lebesgue measurable sets, the inequality had begun its move into the realm of analysis. The last twenty years have seen the Brunn-Minkowski inequality consolidate its role as an analytical tool,

1991 Mathematics Subject Classification. 26D15, 52A40. Key words and phrases. Brunn-Minkowski inequality, Minkowski’s first inequality, Prekopa-Leindler inequality,

Young’s inequality, Brascamp-Lieb inequality, Barthe’s inequality, isoperimetric inequality, Sobolev inequality, entropy power inequality, covariogram, Anderson’s theorem, concave function, concave measure, convex body, mixed volume. Supported in part by NSF Grant DMS 9802388.

2 R. J. GARDNER and a compelling picture (see Figure 1) has emerged of its relations to other analytical inequalities. In an integral version of the Brunn-Minkowski inequality often called the Prekopa-Leindler inequality (12), a reverse form of Holder’s inequality, the geometry seems to have evaporated. Largely through the efforts of Brascamp and Lieb, this can be viewed as a special case of a sharp reverse form (32) of Young’s inequality for convolution norms. A remarkable sharp inequality (36) proved by Barthe, closely related to (32), takes us up to the present time. The modern viewpoint entails an interaction between analysis and convex geometry so potent that whole conferences and books are devoted to “analytical convex geometry” or “convex geometric analysis.”

The main development of this paper includes historical remarks and several detailed proofs that amplify the previous paragraph and show that even the latest developments are accessible to graduate students. Several applications are also discussed at some length. Extensions of the Prekopa-Leindler inequality can be used to obtain concavity properties of probability measures generated by densities of well-known distributions. Such results are related to Anderson’s theorem on multivariate unimodality, an application of the Brunn-Minkowski inequality that in turn is useful in statistics. The entropy power inequality (48) of information theory has a form similar to that of the Brunn-Minkowski inequality. To some extent this is explained by Lieb’s proof that the entropy power inequality is a special case of a sharp form of Young’s inequality (31). This is given in detail along with some brief comments on the role of Fisher information and applications to physics. We come full circle with consequences of the later inequalities in convex geometry. Ball started these rolling with his elegant application of the Brascamp-Lieb inequality (35) to the volume of central sections of the cube and to a reverse isoperimetric inequality (45).

The whole story extends far beyond Figure 1 and the previous paragraph. The final Section 19 is a survey of the many other extensions, analogues, variants, and applications of the Brunn- Minkowski inequality. Essentially the strongest inequality for compact convex sets in the direction of the Brunn-Minkowski inequality is the Aleksandrov-Fenchel inequality (51). Here there is a remarkable link with algebraic geometry: Khovanskii and Teissier independently discovered that the Aleksandrov-Fenchel inequality can be deduced from the Hodge index theorem. Analogues and variants of the Brunn-Minkowski inequality include Borell’s inequality (57) for capacity, employed in the recent solution of the Minkowski problem for capacity; Milman’s reverse Brunn-Minkowski inequality (64), which features prominently in the local theory of Banach spaces; a discrete Brunn- Minkowski inequality (65) due to the author and Gronchi, closely related to a rich area of discrete mathematics, combinatorics, and graph theory concerning discrete isoperimetric inequalities; and inequalities (67), (68) originating in Busemann’s theorem, motivated by his theory of area in Finsler spaces and used in Minkowski geometry and geometric tomography. Around the corner from the Brunn-Minkowski inequality lies a slew of related affine isoperimetric inequalities, such as the Petty projection inequality (62) and Zhang’s affine Sobolev inequality (63), much more powerful than the isoperimetric inequality and the classical Sobolev inequality (24), respectively. There are versions of the Brunn-Minkowski inequality in the sphere, hyperbolic space, Minkowski spacetime, and Gauss space, and there is a Riemannian version of the Prekopa-Leindler inequality, obtained very recently by Cordero-Erausquin, McCann, and Schmuckenschlager. Finally, pointers are given to other applications of the Brunn-Minkowski inequality. Worthy of special mention here is the derivation of logarithmic Sobolev inequalities from the Prekopa-Leindler inequality by Bobkov and Ledoux, and work of Brascamp and Lieb, Borell, McCann, and others on diffusion equations. Measure-preserving convex gradients and transportation of mass, utlilized by McCann

THE BRUNN-MINKOWSKI INEQUALITY OCTOBER 25, 2001 3 in applications to shapes of crystals and interacting gases, were also employed by Barthe in the proof of his inequality.

The reader might share a sense of mystery and excitement. In a sea of mathematics, the Brunn-

Minkowski inequality appears like an octopus, tentacles reaching far and wide, its shape and color changing as it roams from one area to the next. It is quite clear that research opportunities abound. For example, what is the relationship between the Aleksandrov-Fenchel inequality and Barthe’s inequality? Do even stronger inequalities await discovery in the region above Figure 1? Are there any hidden links between the various inequalities in Section 19? Perhaps, as more connections and relations are discovered, an underlying comprehensive theory will surface, one in which the classical Brunn-Minkowski theory represents just one particularly attractive piece of coral in a whole reef. Within geometry, the work of Lutwak and others in developing the dual Brunn-Minkowski and Lp-Brunn-Minkowski theories (see Section 19) strongly suggests that this might well be the case.

An early version of the paper was written to accompany a series of lectures given at the 1999 Workshop on Measure Theory and Real Analysis in Gorizia, Italy. I am very grateful to Franck Barthe, Apostolos Giannopoulos, Paolo Gronchi, Peter Gruber, Daniel Hug, Elliott Lieb, Robert McCann, Rolf Schneider, Bela Uhrin, and Gaoyong Zhang for their extensive comments on previous versions of this paper, as well as to many others who provided information and references.

2. A first step

An old saying has it that even a journey of a thousand miles must begin with a single step. Ours will be the following easy result (see Section 3 for definitions and notation).

Theorem 2.1. (Brunn-Minkowski inequality in R.) Let 0 < λ < 1 and let X and Y be nonempty bounded measurable sets in R such that (1 − λ)X + λY is also measurable. Then

Proof. Suppose that X and Y are compact sets. It is straightforward to prove that X +Y is also compact. Since the measures do not change, we can translate X and Y so that X ∩ Y = {o}, X ⊂ {x : x ≤ 0}, and Y ⊂ {x : x ≥ 0}. Then X +Y ⊃ X ∪Y, so

If we replace X by (1−λ)X and Y by λY , we obtain (2) for compact X and Y . The general case follows easily by approximation from within by compact sets. ¤

Simple though it is, Theorem 2.1 already raises two important matters. Firstly, observe that it was enough to prove the theorem when the factors (1 − λ) and λ are omitted. This is due to the positive homogeneity (of degree 1) of Lebesgue measure in R:

V1(rX) = rV1(X) for r ≥ 0. In fact, this property allows these factors to be replaced by arbitrary nonnegative real numbers. For reasons that will become clear, it will be convenient for most of the paper to incorporate the factors (1 − λ) and λ.

Secondly, the set (1−λ)X +λY may not be measurable, even when X and Y are measurable.

We discuss this point in more detail in Section 9.

The assumption in Theorem 2.1 and its n-dimensional forms, Theorem 5.1 and Corollary 5.3 below, that the sets are bounded is easily removed and is retained simply for convenience.

4 R. J. GARDNER

Prekopa-Leindler (12) Holder (1)

General Brunn-Minkowski (14)

Aleksandrov-Fenchel (51) Barthe (36)

Brascamp-Lieb (35)

Reverse Young (32) Young (31)

Brunn-Minkowski for C1 domains Sobolev for C1 functions (24)

Brunn-Minkowski for convex bodies (1) Minkowski’s first for convex bodies (20)

Isoperimetric for C1 domains Isoperimetric for convex bodies (21)

Entropy power (48).' .

Figure 1. Relations between inequalities labeled as in the text.

THE BRUNN-MINKOWSKI INEQUALITY OCTOBER 25, 2001 5

3. A few preliminaries

We denote the origin, unit sphere, and closed unit ball in n-dimensional Euclidean space Rn by o, Sn−1, and B, respectively. The Euclidean scalar product of x and y will be written x · y, and ‖x‖ denotes the Euclidean norm of x. If u ∈ Sn−1, then u⊥ is the hyperplane containing o and orthogonal to u.

Hausdorff measure in Rn. Then spherical Lebesgue measure in Sn−1 can be identified with Vn−1 in Sn−1. In this paper dx will denote integration with respect to Vk for the appropriate k and integration over Sn−1 with respect to Vn−1 will be denoted by du.

The term “measurable” applied to a set in Rn will mean Vn-measurable unless stated otherwise.

If X is a compact set in Rn with nonempty interior, we often write V (X) = Vn(X) for its volume. We shall do this in particular when X is a convex body, a compact convex set with nonempty interior. We also write κn = V (B). In geometry, it is customary to use the term volume, more generally, to mean the k-dimensional Lebesgue measure of a k-dimensional compact body X (equal to the closure of its relative interior), i.e. to write V (X) = Vk(X) in this case. Let X and Y be sets in Rn. We define their vector or Minkowski sum by

If r > 0, then rX is the dilatation of X with factor r, and if r < 0, it is the reflection of this dilatation in the origin. If 0 < λ < 1, the set (1 − λ)X + λY is called a convex combination of X and Y . Minkowski’s definition of the surface area S(M) of a suitable set M in Rn is In this paper we will use this definition when M is a convex body or a compact domain with piecewise C1 boundary. A function f on Rn is concave on a convex set C if for all x,y ∈ C and 0 < λ < 1, and a function f is convex if −f is concave. A nonnegative function f is log concave if log f is concave. Since the latter condition is equivalent to the arithmetic-geometric mean inequality implies that each concave function is log concave. If f is a nonnegative measurable function on Rn and t ≥ 0, the level set L(f,t) is defined by

By Fubini’s theorem,∫ 1dxdt =

6 R. J. GARDNER If E is a set, 1E denotes the characteristic function of E. The formula representation of f.

4. The Prekopa-Leindler inequality

(Parte 1 de 10)