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.
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.
Reviewer: Laszlo Leindler (Szeged)
MathOverflow Questions:
Log-concavity of areas of level setsProduct of concave functions and harmonic mean
MSC:
| 26D15 | Inequalities for sums, series and integrals |
| 52A40 | Inequalities and extremum problems involving convexity in convex geometry |
Keywords:
Brunn-Minkowski inequality; Minkowski’s first inequality; Prékopa-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 volumeReferences:
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