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Degeneracy criterion for a convex polyhedron. (English. Russian original) Zbl 1264.53065

Russ. Math. Surv. 67, No. 5, 951-953 (2012); translation from Usp. Mat. Nauk. 67, No. 5, 175-176 (2012).
An “abstract sphere” is a finite union of convex Euclidean polygons glued together pairwise along common edges of equal length to produce a surface homeomorphic to \(\mathbb S^2.\) For such a surface the well-known Euler formula, relating the number of vertices (\(v\)), edges (\(e\)) and faces (\(f\)) holds, namely: \(v - e + f = 2\). The curvature of such a surface at a chosen vertex is defined by \(\omega = 2\pi - \Sigma\, \alpha, \) which is non-negative when the sum of the planar angles \(\Sigma\, \alpha\) at that vertex is \( \leq 2\pi.\) The total curvature \(\Omega = \Sigma\, \omega\) of an abstract sphere equals \(4 \pi\) by Euler’s formula. In [Convex polyhedra. Konvexe Polyeder. Berlin: Akademie-Verlag (2005; Zbl 0079.16303)] A. D. Alexandrov proved that if the curvature of an abstract 2-dimensional sphere at its vertices is positive then there exists a convex polyhedron in \(\mathbb R^3\) whose surface is isometric to this abstract sphere. Such apolyhedron is allowed to be also degenerate (with zero volume, contained in \(\mathbb R^2\)), in which case its surface consists of two equal convex polygons with common boundary.
In this paper the author provides a criterion to determine when a given abstract sphere satisfying the hypotheses of the Alexandrov theorem above has the associated convex polyhedron (the existence of which is guaranteed by that theorem) degenerate. He then uses that criterion and Alexandrov’s theorem to give an alternative proof of the following result by A. A. Zil’berberg [Usp. Mat. Nauk 17, No. 4(106), 119–126 (1962; Zbl 0113.16103)]. For any set of \(n \geq 4\) real numbers \(k_i, \, 1 \leq i \leq n,\) satisfying the conditions \(0 < k_1 \leq k_2 \leq \cdots \leq k_n < 2\pi, \, \) \(k_1 + k_2 + \cdots + k_n = 4\pi,\) there exists a non-degenerate convex polyhedron in \(\mathbb R^3\) with \(n\) vertices at which the curvatures of the boundary surface are respectively equal to \(k_1, k_2, \cdots , k_n.\)

MSC:

53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
52A15 Convex sets in \(3\) dimensions (including convex surfaces)
52B10 Three-dimensional polytopes
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