HYPERBOLIC VIRTUAL POLYTOPES
Introductive and popular papers, lecture notes 3D models and movies All available papers on hyperbolic polytopes and related topics More Russian
The starting point of this theory was a paper by A.D. Alexandrov which has led to the following conjecture.
Let K be a smooth convex body in 3D Euclidean space. If there exists a constant ׁ which separates (non-strictly) the principal curvatures at each point of the boundary of K, then K is a ball of radius C.
The first counterexample to the conjecture was constructed by Yves Martinez-Maure in 2001. Nearly at the same time, there appeared a paper by A.V. Pogorelov with an erroneous proof of the conjecture. Being intrigued by the evident contradiction, Viktor Alexandrov asked me to clarify the situation. To understand what is going on, I developed the theory of hyperbolic virtual polytopes based on technique of virtual polytopes developed by ְ. Pukhlikov and ְ. Khovanskii On the one hand, the counterexample of Y. Martinez-Maure is correct (and therefore, A.V. Pogorelow was wrong). Moreover, it turned out that there exist multiple counterexamples to the conjecture. On the other hand, A.V. Pogorelov's method has led to an interesting theorem on inflection arches of saddle spheres.
First 3D images of hyperbolic virtual polytopes (see 3D gallery) were produced by my PhD student   Marina Knyazeva .
Besides, there exists a natural relationship with the theory of עומנטוי pseudotriangulations .