Rigidity and flexibility of virtual polytopes
A virtual polytope is the Minkowski difference of two convex compact polytopes in \(\mathbb R^n\). The notion of a virtual polytope was independently introduced by several authors [see, e.g., \textit{P. McMullen}, Adv. Math. 78, 76-130 (1989; Zbl 0686.52005)] or \textit{A. V. Pukhlikov} and \textit{A. G. Khovanskij}, St. Petersbg. Math. J. 4, 337-356 (1993; Zbl 0791.52010)]. Virtual polytopes form an abelian group with respect to the Minkowski addition \(\otimes\). It is possible to introduce mutually consistent notions of face, edge, volume, mixed volume, support function, and fan for virtual polytopes. The volume \(V(\lambda K\otimes \mu L)\) of a linear combination of virtual polytopes \(K\) and \(L\) is a polynomial in \(\lambda\) and \(\mu\). A virtual polytope \(K\) is said to be flexible if there exists a nontrivial (i.e., not generated by a motion) continuous family of virtual polytopes \(K_t\), \(t\in [0,1]\), \(K_0=K\), such that every two \(K_t\) are combinatorially equivalent and their correspondent faces can be obtained by a motion one from another. The author proves that a virtual polytope in \(\mathbb R^3\) with a convex fan is not flexible, which is a refinement of the rigidity theorem for herissons obtained in [\textit{L. Rodriguez} and \textit{H. Rosenberg}, Comment. Math. Helv. 75, 478-503 (2000; Zbl 0968.52018)]. She also gives examples of flexible virtual polytopes which are similar to Bricard's flexible octahedra.
scientific article; zbMATH DE number 1978729
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