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2003 | 1 | 2 | 157-168
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Rigidity and flexibility of virtual polytopes

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Języki publikacji
EN
Abstrakty
EN
All 3-dimensional convex polytopes are known to be rigid. Still their Minkowski differences (virtual polytopes) can be flexible with any finite freedom degree. We derive some sufficient rigidity conditions for virtual polytopes and present some examples of flexible ones. For example, Bricard's first and second flexible octahedra can be supplied by the structure of a virtual polytope.
Słowa kluczowe
Wydawca
Czasopismo
Rocznik
Tom
1
Numer
2
Strony
157-168
Opis fizyczny
Daty
wydano
2003-06-01
online
2003-06-01
Twórcy
autor
Bibliografia
  • [1] A.D. Alexandrov: Konvexe Polyeder, Berlin, Akademie-Verlag (1958).
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  • [4] V. Danilov: “The geometry of toric varieties”, Russian Math. Surveys, Vol. 33, (1978), pp. 97–154. http://dx.doi.org/10.1070/RM1978v033n02ABEH002305
  • [5] A. Khovanskii, A. Pukhlikov: “Finitely additive measures of virtual polytopes”, St. Petersburg Math. J., Vol. 4, (1993), pp. 337–356.
  • [6] K. Leichtweiss: Konvexe Mengen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1980.
  • [7] P. McMullen: “The polytope algebra”, Adv. Math., Vol. 78, (1989), pp. 76–130. http://dx.doi.org/10.1016/0001-8708(89)90029-7
  • [8] P. McMullen: “On simple polytopes”, Invent. Math., Vol. 113, (1993), pp. 19–111. http://dx.doi.org/10.1007/BF01244313
  • [9] Y. Martinez-Maure: “De nouvelles inégalités géométriques pour les hérissons”, Arch. Math., Vol. 72, (1999), pp. 444–453. http://dx.doi.org/10.1007/s000130050354
  • [10] Y. Martinez-Maure: “Contre-exemple à une caractérisation conjecturée de la sphère”, C.R. Acad. Sci. Paris, Vol 332, (2001), pp. 41–44.
  • [11] G. Panina: “Mixed volumes for non-convex bodies”, Isv. Akad. Nauk Armenii, Matematika, Vol. 28, (1993), pp. 51–59.
  • [12] G. Panina: “The structure of virtual polytope group related to cylinders subgroups”, St. Petersburg Math. J., Vol. 13, (2001), pp. 471–484.
  • [13] G. Panina: “Virtual polytopes and some classical problems”, St. Petersburg Math. J., Vol. 14, (2002), pp. 152–170.
  • [14] H. Radström: “An embedding theorem for spaces of convex sets”, Proc. AMS, Vol. 3, (1952), pp. 165–169. http://dx.doi.org/10.2307/2032477
  • [15] L. Rodriguez and H. Rosenberg: “Rigidity of certain polyhedra in ℝ3”, Comment. Math. Helv., Vol. 75, (2000), pp. 478–503. http://dx.doi.org/10.1007/s000140050137
  • [16] I. Sabitov: “The local theory of bendings of surfaces”, in: Encyclopaedia of Math. Sc., Vol. 48, (1992), Geometry 3, pp. 179–250.
  • [17] O.Ya. Viro: “Some integral calculus based on Euler characteristic”, Topology and Geometry-Rokhlin Seminar, Lecture Notes in Math., 1346, Springer-Verlag, Berlin-New York, 1988, pp. 127–138.
  • [18] O.Ya. Viro: “Some integral calculus based on Euler characteristic”, Topology and Geometry-Rokhlin Seminar, Lecture Notes in Math., 1346, Springer-Verlag, Berlin-New York, 1988, pp. 127–138.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_BF02476005
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