1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

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Convex bodies : the Brunn-Minkowski theory

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 Rn, 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.

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The Brunn-Minkowski inequality

We consider approximations of a smooth convex body by inscribed and circumscribed convex polytopes as the number of vertices, resp. facets tends to infinity. The measure of deviation used is the dierence of the mean width of the convex body and the approximating polytopes. The following results are obtained. (i) An asymptotic formula for best approximation. (ii) Upper and lower estimates for step- by-step approximation in terms of the so-called dispersion. (iii) For a sequence of best approximating inscribed polytopes the sequence of vertex sets is uniformly distributed in the boundary of the convex body where the density is specified explicitly.

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Asymptotic estimates for best and stepwise approximation of convex bodies III

Mixed projection bodies are related to ordinary projection bodies (zonoids) in the same way that mixed volumes are related to ordinary volume. Analogs of the classical inequalities from the Brunn-Minkowski Theory (such as the Minkowski, Brunn-Minkowski, and Aleksandrov-Fenchel inequalities) are developed for projection and mixed projection bodies

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Inequalities for mixed projection bodies

Publisher Summary This chapter focuses on zonoids and generalizations and discusses analytic aspects of centrally symmetric convex bodies Zonoids are limits of zonotopes in the Hausdorff metric Zonoids occur as images of a natural operation—namely, as projection bodies Zonoids easily allow extending some analytic notions, properties, and formulae to more general convex sets (generalized zonoids) or even to all centrally symmetric convex bodies The characterization of surface area measures of convex bodies shows that centered zonoids are precisely the projection bodies of convex bodies A centrally symmetric polytope with nonsymmetric faces, for example, an octahedron, is not a zonoid The only polytopal generalized zonoids are the zonotopes The generalized zonoids are dense in the centrally symmetric bodies Any sufficiently smooth centrally symmetric body should be a generalized zonoid There is a hierarchy of centrally symmetric convex bodies corresponding to the nature of the generating distribution Zonotopes are the bodies whose generating distribution is an atomic measure Zonoids are those for which it is a positive measure, and the generalized zonoids correspond to the case of signed measures

Zonoids and Generalisations

Most geometric inequalities concerning convex bodies have the property that the occurrence of the equality sign characterizes geometrically significant objects, like balls, ellipsoids or simplexes. This fact suggests the following stability problem: If for some convex body the given inequality is satisfied so that it is not very different from an equality what can be said about the deviation of the body from these objects? Problems of this kind appear already in the work of Minkowski and Bonnesen but have been investigated more systematically only in recent times. The present article is a survey of stability results for geometric inequalities and provides pertinent references. Because of space limitations it was not possible to include complete proofs but occasionally some indications regarding the methods of proof will be provided.

Stability of Geometric Inequalities

Stability Estimates for some Geometric Inequalities

Geometric inequalities, like the isoperimetric inequality, have been a subject of intensive research for a long time. More recently, some properties of these inequalities, which may be called stability properties, have been investigated. Roughly speaking, these investigations concern the geometric implications if the inequalities are in a certain sense close to equalities. The present article is primarily an exposition of various stability results concerning inequalities for plane convex sets, including a discussion of the concepts and methods which are of importance in this area. Moreover, several new results and proofs are presented. 1. Introduction. Let RI denote n-dimensional Eucidean space. A bounded con- vex subset of R" will be called an n-dimensional convex body if it is closed and has interior points. We let qpn denote the class of all n-dimensional convex bodies. Two-dimensional convex bodies will be called convex domains. We consider primarily inequalities of the form

Stability properties of geometric inequalities

The motivation for this work comes from a result of Minkowski. He showed that if a three-dimensional convex body has the property that all its projections have the same perimeter, then the original body has constant width. Our objective was to extend this to a stability result and not to restrict ourselves to dimension three. The result we obtained shows that if two centrally symmetric bodies have projections which all have approximately the same mean width, then the two bodies are approximately the same up to translation. This is, in effect, a continuity result for the inverse of the spherical Radon transform. It is closely related to recent three-dimensional results of Campi and to work of Bourgain and Lindenstrauss, who consider the volumes of projections rather than their mean widths. The techniques we employ are drawn from the theory of spherical harmonics and from the theory of mixed volumes.

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H. Groemer

Papers

Stability of Geometric Inequalities

Stability Estimates for some Geometric Inequalities

Stability properties of geometric inequalities

Stability results for first order projection bodies

On Finite Classes of Convex Sets that Permit Space Coverings