A. D. Alexandrov's uniqueness theorem for convex polytopes and its refinements (Q2473577)

The celebrated Alexandrov theorem states that if no parallel faces of two three-dimensional convex polytopes can be placed strictly one into another via a translation, then the polytopes are translates of one another. Similar uniqueness statement for smooth convex surfaces conjectured by Alexandrov turns out to be wrong, for counterexamples [see \textit{Y. Martinez-Maure}, C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 1, 41--44 (2001; Zbl 1008.53002); \textit{G. Panina}, Cent. Eur. J. Math. 4, No. 2, 270--293 (2006; Zbl 1107.52002), Adv. Geom. 5, No. 2, 301--317 (2005; Zbl 1077.52003)]. The author presents two refinements of the Alexandrov theorem, obtained with the help of hyperbolic virtual polytopes. On one hand, it is demonstrated that, given two convex polytopes, if for any pair of parallel faces (i) there exists at most one translation placing the face of the first polytope strictly inside the face of the second one and (ii) there exists no translation placing the face of the second polytope strictly inside the face of the first one, then the polytopes are translates of one another. On the other hand, the author constructs an example of two different convex polytopes such that, for each pair of their parallel faces, there exists at most one translation placing one of the faces into another; the constructed polytopes have 56 faces, an open problem is to present more trivial examples.