The paper gives an illustrated introduction to the theory of hyperbolic virtual polytopes and related counterexamples to A.D. Alexandrov’s conjecture.
@article{bwmeta1.element.doi-10_2478_s11533-008-0020-1, author = {Marina Knyazeva and Gaiane Panina}, title = {An illustrated theory of hyperbolic virtual polytopes}, journal = {Open Mathematics}, volume = {6}, year = {2008}, pages = {204-217}, zbl = {1145.52004}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0020-1} }
Marina Knyazeva; Gaiane Panina. An illustrated theory of hyperbolic virtual polytopes. Open Mathematics, Tome 6 (2008) pp. 204-217. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0020-1/
[1] Alexandrov A.D., Sur les théoremes d’unicité pour les surfaces fermées, C. R. (Dokl.) Acad. Sci. URSS, 1939, 22, 99–102 | Zbl 65.0828.03
[2] Martinez-Maure Y., A counterexample to a conjectured characterization of the sphere, C. R. Acad. Sci. Paris Sér. I Math., 2001, 332, 41–44 (in French) | Zbl 1008.53002
[3] Martinez-Maure Y., Hedgehog theory and polytopes, C. R. Math. Acad. Sci. Paris, 2003, 336, 241–244 (in French) | Zbl 1053.52009
[4] Panina G., Isotopy problems for saddle surfaces, preprint available at ESI preprints http://www.esi.ac.at/Preprint-shadows/esi1796.html
[5] Panina G., New counterexamples to A.D. Alexandrov’s hypothesis, Adv. Geom., 2005, 5, 301–317 http://dx.doi.org/10.1515/advg.2005.5.2.301 | Zbl 1077.52003
[6] Panina G., On hyperbolic virtual polytopes and hyperbolic fans, Cent. Eur. J. Math., 2006, 4, 270–293 http://dx.doi.org/10.2478/s11533-006-0006-9 | Zbl 1107.52002
[7] Pukhlikov A.V., Khovanskiĭ A.G., Finitely additive measures of virtual polyhedra, St. Petersburg Math. J., 1993, 4, 337–356