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Valuations on convex bodies. (English) Zbl 0534.52001
Convexity and its applications, Collect. Surv., 170-247 (1983).
[For the entire collection see Zbl 0512.00020.]
Valuations play an important, unifying and clarifying, role in almost all areas of convexity, and their systematic study has lead to considerable progress in many of these areas. The authors, who both have made important contributions to the theory of valuations, have succeeded in writing a survey which is attractive for a general public, and yet provides a broad and solid basis for future research.
Chapter 1 of their work deals with the continuous valuations arising in Minkowski’s theory of convex bodies, such as the volume, the mixed volumes, the quermass integrals, as well as their vector valued and their local analogues. It describes recent attempts to extend appropriate parts of this theory to more general sets and to other homogeneous spaces. Chapter 2 is devoted to Hilbert’s third problem and thus to the question of equidissectability for pairs of polyhedra. This leads to the theory of discrete valuations on the class of all Euclidean, spherical or hyperbolic polyhedra, with certain invariance properties. The authors describe the highlights of this theory, arising from the work of Dehn, Hadwiger, Sydler, Jessen, Thorup and Sah, as well as some of its fascinating open problems.
A relatively recent development of the theory of valuations is the subject of chapter 3. It establishes, in the context of combinatorial geometry, a number of identities similar to the Euler formula, for quite arbitrary valuations. These identities have interesting applications to the study of lattice polytopes. The final chapter 4 presents several old and new characterization theorems for the classical valuations, and also contains a number of interesting open problems.
Reviewer: P.Mani

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
11H06 Lattices and convex bodies (number-theoretic aspects)
52Bxx Polytopes and polyhedra
52A55 Spherical and hyperbolic convexity
52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry