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Rigidity. (English) Zbl 0788.52001

Gruber, P. M. (ed.) et al., Handbook of convex geometry. Volume A. Amsterdam: North-Holland. 223-271 (1993).
This article is a survey which is representative for the high standard of the excellent articles contained in this handbook. [P. M. Gruber and J. M. Wills, Handbook of convex geometry. Vol. A. (1993; Zbl 0777.52001).]
Rigidity is a classical subject going back to the beginning of geometry. The author gives a motivating introduction and a comprehensive survey of the different facets of this subject.
The introduction is combined with a discussion of the famous Cauchy’s Theorem on the rigidity of convex polyhedra and its proof. This is extended to smooth analogues of this theorem and to the Aleksandrov- Pogorelov theory. Then the theory of rigidity of bar frameworks and more generally of tensegrity frameworks is presented. The discussion concerns different types of rigidity as well as technical items like the projective invariance or the rigidity map. The results of this theory are used for the study of infinitesimal and static rigidity related to surfaces. The subject of this section are Grünbaum and Shephard’s conjectures, Aleksandrov’s theory, Dehn’s theorem, Maxwell-Cremona theory and spider webs, problems in higher dimensions etc. The final section of this article is devoted to second-order rigidity and pre-stress stability of tensegrity frameworks and polyhedral surfaces.
Clearly, it is not possible to mention all of the interesting results covered by this article in detail in this review. The article will be a good basis for anyone who wants to become acquainted with the fascinating achievements in rigidity theory and to start with his own investigations in this area.
For the entire collection see [Zbl 0777.52001].

MSC:

52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry
52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
53A05 Surfaces in Euclidean and related spaces
52B10 Three-dimensional polytopes

Citations:

Zbl 0777.52001
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