Convex bodies: the Brunn-Minkowski theory.

*(English)*Zbl 0798.52001
Encyclopedia of Mathematics and Its Applications. 44. Cambridge: Cambridge University Press. xiii, 490 p. (1993).

Classical convex geometry emerged from the results of Brunn and Minkowski at the end of the last century. These include the celebrated Brunn- Minkowski theorem and its consequences such as Minkowski’s inequalities for mixed volumes. A first period of fruitful research was summarized by T. Bonnesen and W. Fenchel [Theorie der konvexen Körper, Springer, Berlin (1934; Zbl 0008.07708)]. This monograph was the standard reference for several decades, even though many interesting developments took place subsequently (the area measures of Alexandrov, Fenchel and Jessen; the Alexandrov-Fenchel inequalities; Hadwiger’s axiomatic characterization of quermassintegrals; Federer’s curvature measures).

In recent years, convexity theory has been of increasing interest with connections to various other disciplines, and the Brunn-Minkowski theory has maintained its place at the centre of the field. The more recent developments include a solution of the Christoffel problem, local formulae of integral geometry, the theory of valuations, new inequalities for various functionals, zonoids and their applications in stochastic geometry, stability results improving classical inequalities, and, as one of the major challenges, the equality case in the Alexandrov-Fenchel inequality has been attacked, although the final solution here is still to come.

Over time, the necessity of a book, which complements the volume of Bonnesen and Fenchel and includes the most important of these new developments, became clear. The monograph under review satisfies this need in an impressive manner. It is written by an author who has himself made important contributions to nearly all aspects of the theory and whose work has initiated many of the recent developments.

The book follows a carefully constructed path leading from the basics of convexity theory to the Aleksandrov-Fenchel inequalities, its variants and applications. The first chapter contains the fundamental notions and properties of convex sets and functions in \({\mathbf E}^ n\), as they are found in most books dealing with convexity (metric projection, support and separation properties, extremal representations, regularity properties and characterizations of convex functions, duality, support functions, Hausdorff metric). The presentation here, as in the later parts, is clear and concise, providing the reader with the necessary arguments and giving more details, where it is appropriate (for example, three different proofs are given for the fundamental relation between convex bodies and sublinear functions, each of which is of special interest).

The second chapter discusses the boundary structure of convex bodies. Besides classical material about faces and singularities of convex sets, the theorem of Ewald-Larman-Rogers is presented (concerning the measure of the set of directions of segments in the boundary of a convex body). It follows a section of polytopes, which are treated here mostly as tools for later results on general convex bodies. For example, the simultaneous approximation of \(m\) convex bodies \(K_ 1,\dots, K_ m\) by simple strongly isomorphic polytopes is proved here, which is a key to the later proof of the Alexandrov-Fenchel inequality. As another class of special interest, smooth bodies and their curvature properties are then discussed, and the chapter ends with some results on the ‘generic’ boundary behaviour of convex bodies, that is, on properties which are fulfilled by almost all convex bodies (in the Baire category sense).

Chapter 3 is devoted to Minkowski addition. Summands, indecomposability, approximation by smooth bodies (using a convolution argument), additive mappings and valuations are some of the key words here. Zonoids and their generalizations are treated as examples of classes which are closed under addition. Here, spherical harmonics are used to give short proofs of the injectivity and surjectivity properties of the underlying spherical transform.

In the next chapter, local Steiner formulae are used to introduce the curvature measures and area measures of a convex body and their global analogs, the quermassintegrals (intrinsic volumes). The area measures of order one are treated separately and the solution due to Firey and Berg of the Christoffel problem is given without proof. The additive extensions to the convex ring then prepare the ground for the integral geometric formulae which are presented in their local version, for curvature measures (resp. area measures). These integral geometric results are used in the final section to obtain some support properties of area measures.

Chapter 5 introduces the basic notions for the remainder of the book, the mixed volumes and their local counterparts, the mixed area measures. The technical difficulties which often occur when proving the elementary properties of mixed volumes are circumvented by using approximation with strongly isomorphic polytopes and an ‘inversion formula’, which expresses the mixed volume as an alternating sum of ordinary volumes (of sum sets). A short section on the question of extending mixed volumes is followed by a collection of special formulae (for smooth bodies and zonoids, as well as integral geometric formulae) and by a section on moment vectors and curvature centroids.

Chapter 6, which is the heart of the book, starts with the Brunn- Minkowski theorem, for which one proof is given and another one is sketched. The Minkowski and isoperimetric inequalities follow and then the Alexandrov-Fenchel inequality is proved. Consequences and improvements are discussed in section 6.4 (including the general Brunn- Minkowski theorem). Section 6.5 on generalized parallel bodies contains the proof of a strengthened version of the Minkowski inequalities (due to Diskant) stated earlier, and it provides the tools for results on the equality case in section 6.6. This section 6.6 is quite substantial, it presents a number of uniqueness results and their more recent stability versions, then conjectures concerning the equality case in the Alexandrov-Fenchel inequality are formulated and the positive results which exist in these directions are discussed (some without proofs). After a section on linear improvements of the quadratic inequalities, a few variants of the classical inequalities are collected without proof (for dual quermassintegrals and harmonic quermassintegrals).

The last chapter contains selected applications of the Brunn-Minkowski theory. Minkowski’s existence theorem, uniqueness results for area measures, and a number of further inequalities for convex bodies (inequality for the difference body, Petty’s projection inequality, Busemann’s intersection inequality, the Busemann-Petty centroid inequality, the Blaschke-Santalò inequality) are the topics discussed here, some without proofs.

The book finishes with an appendix on spherical harmonics and a very helpful and extensive bibliography. The material presented in detail in the seven chapters is complemented by notes and comments at the end of each section. They offer an invaluable variety of additional results, give cross references and also provide the reader with helpful informations on the history and priority of the main results. This volume will surely be the standard reference for the future work in the theory of convex bodies.

In recent years, convexity theory has been of increasing interest with connections to various other disciplines, and the Brunn-Minkowski theory has maintained its place at the centre of the field. The more recent developments include a solution of the Christoffel problem, local formulae of integral geometry, the theory of valuations, new inequalities for various functionals, zonoids and their applications in stochastic geometry, stability results improving classical inequalities, and, as one of the major challenges, the equality case in the Alexandrov-Fenchel inequality has been attacked, although the final solution here is still to come.

Over time, the necessity of a book, which complements the volume of Bonnesen and Fenchel and includes the most important of these new developments, became clear. The monograph under review satisfies this need in an impressive manner. It is written by an author who has himself made important contributions to nearly all aspects of the theory and whose work has initiated many of the recent developments.

The book follows a carefully constructed path leading from the basics of convexity theory to the Aleksandrov-Fenchel inequalities, its variants and applications. The first chapter contains the fundamental notions and properties of convex sets and functions in \({\mathbf E}^ n\), as they are found in most books dealing with convexity (metric projection, support and separation properties, extremal representations, regularity properties and characterizations of convex functions, duality, support functions, Hausdorff metric). The presentation here, as in the later parts, is clear and concise, providing the reader with the necessary arguments and giving more details, where it is appropriate (for example, three different proofs are given for the fundamental relation between convex bodies and sublinear functions, each of which is of special interest).

The second chapter discusses the boundary structure of convex bodies. Besides classical material about faces and singularities of convex sets, the theorem of Ewald-Larman-Rogers is presented (concerning the measure of the set of directions of segments in the boundary of a convex body). It follows a section of polytopes, which are treated here mostly as tools for later results on general convex bodies. For example, the simultaneous approximation of \(m\) convex bodies \(K_ 1,\dots, K_ m\) by simple strongly isomorphic polytopes is proved here, which is a key to the later proof of the Alexandrov-Fenchel inequality. As another class of special interest, smooth bodies and their curvature properties are then discussed, and the chapter ends with some results on the ‘generic’ boundary behaviour of convex bodies, that is, on properties which are fulfilled by almost all convex bodies (in the Baire category sense).

Chapter 3 is devoted to Minkowski addition. Summands, indecomposability, approximation by smooth bodies (using a convolution argument), additive mappings and valuations are some of the key words here. Zonoids and their generalizations are treated as examples of classes which are closed under addition. Here, spherical harmonics are used to give short proofs of the injectivity and surjectivity properties of the underlying spherical transform.

In the next chapter, local Steiner formulae are used to introduce the curvature measures and area measures of a convex body and their global analogs, the quermassintegrals (intrinsic volumes). The area measures of order one are treated separately and the solution due to Firey and Berg of the Christoffel problem is given without proof. The additive extensions to the convex ring then prepare the ground for the integral geometric formulae which are presented in their local version, for curvature measures (resp. area measures). These integral geometric results are used in the final section to obtain some support properties of area measures.

Chapter 5 introduces the basic notions for the remainder of the book, the mixed volumes and their local counterparts, the mixed area measures. The technical difficulties which often occur when proving the elementary properties of mixed volumes are circumvented by using approximation with strongly isomorphic polytopes and an ‘inversion formula’, which expresses the mixed volume as an alternating sum of ordinary volumes (of sum sets). A short section on the question of extending mixed volumes is followed by a collection of special formulae (for smooth bodies and zonoids, as well as integral geometric formulae) and by a section on moment vectors and curvature centroids.

Chapter 6, which is the heart of the book, starts with the Brunn- Minkowski theorem, for which one proof is given and another one is sketched. The Minkowski and isoperimetric inequalities follow and then the Alexandrov-Fenchel inequality is proved. Consequences and improvements are discussed in section 6.4 (including the general Brunn- Minkowski theorem). Section 6.5 on generalized parallel bodies contains the proof of a strengthened version of the Minkowski inequalities (due to Diskant) stated earlier, and it provides the tools for results on the equality case in section 6.6. This section 6.6 is quite substantial, it presents a number of uniqueness results and their more recent stability versions, then conjectures concerning the equality case in the Alexandrov-Fenchel inequality are formulated and the positive results which exist in these directions are discussed (some without proofs). After a section on linear improvements of the quadratic inequalities, a few variants of the classical inequalities are collected without proof (for dual quermassintegrals and harmonic quermassintegrals).

The last chapter contains selected applications of the Brunn-Minkowski theory. Minkowski’s existence theorem, uniqueness results for area measures, and a number of further inequalities for convex bodies (inequality for the difference body, Petty’s projection inequality, Busemann’s intersection inequality, the Busemann-Petty centroid inequality, the Blaschke-Santalò inequality) are the topics discussed here, some without proofs.

The book finishes with an appendix on spherical harmonics and a very helpful and extensive bibliography. The material presented in detail in the seven chapters is complemented by notes and comments at the end of each section. They offer an invaluable variety of additional results, give cross references and also provide the reader with helpful informations on the history and priority of the main results. This volume will surely be the standard reference for the future work in the theory of convex bodies.

Reviewer: W.Weil (Karlsruhe)

##### MSC:

52-02 | Research exposition (monographs, survey articles) pertaining to convex and discrete geometry |

52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |

52A39 | Mixed volumes and related topics in convex geometry |

52A40 | Inequalities and extremum problems involving convexity in convex geometry |

52A22 | Random convex sets and integral geometry (aspects of convex geometry) |

33C55 | Spherical harmonics |

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