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An analogue of the Hodge-Riemann relations for simple convex polytopes. (English. Russian original) Zbl 0944.52005
Russ. Math. Surv. 54, No. 2, 381-426 (1999); translation from Usp. Mat. Nauk 54, No. 2, 113-162 (1999).
In a number of respects, this is a somewhat curious paper. While it presents the polytope algebra, described originally in the reviewer’s paper [Adv. Math. 78, No. 1, 76-130 (1989; Zbl 0686.52005)], in a way based on work of Pukhlikov and Khovanskii, and therefore starting from the volume form, the bulk of the paper consists of an almost direct translation (into Russian and different mathematical language – and then, of course, retranslation into English) of appropriate parts of the reviewer’s paper [Invent. Math. 113, No. 2, 419-444 (1993; Zbl 0803.52007)], which gave an elementary proof (that is, not using algebraic geometry) of the \(g\)-theorem, characterizing the possible numbers of faces of simple convex polytopes. The alternative approach is of interest, and does simplify some parts of the argument. However, the author has evidently not seen two papers by the reviewer [Discrete Comput. Geom. 15, No. 4, 363-388 (1996; Zbl 0849.52011)] and C. W. Lee [Discrete Comput. Geom. 15, No. 4, 389-421 (1996; Zbl 0856.52009)]. The first provides (in the weight algebra) a conceptually even simpler framework for the polytope algebra, and the second details the connexions between it and the dual notion of stress, whose description also relies on the volume form.

52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
52B45 Dissections and valuations (Hilbert’s third problem, etc.)
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