×

Some problems on polyhedra. (English) Zbl 0618.52004

Starting with a problem of Jacob Steiner the authors give a list of 16 open problems on convex polyhedra in Euclidean 3-space. All problems are more or less related to isomorphism (i.e. combinatorial equivalence) and duality of polyhedra.
Reviewer: J.M.Wills

MSC:

52Bxx Polytopes and polyhedra
Full Text: DOI

References:

[1] A. Brøndsted, An Introduction to Convex Polytopes. Springer-Verlag, New York 1983. · Zbl 0509.52001
[2] M. Brückner, Vielecke und Vielflache. Teubner, Leipzig 1900.
[3] H. M. Cundy and A. P. Rollett, Mathematical Models. Second ed. Clarendon Press, Oxford 1961. · Zbl 0095.38001
[4] B. Grünbaum, On Steinitz’s theorem about non-inscribable polyhedra.Ned. Akad. Wetenschap. Proc. Ser. A, 66(1983), 452-455. · Zbl 0115.15005
[5] B. Grünbaum, Convex Polytopes. Wiley, New York 1967. · Zbl 0163.16603
[6] B. Grünbaum, Polytopes, graphs, and complexes.Bull. Amer. Math. Soc. 76(1970), 1131-1201. · Zbl 0211.25001 · doi:10.1090/S0002-9904-1970-12601-5
[7] B. Grünbaum and E. Jucovi?, On non-inscribable polytopes.Czechoslovak Math. J. 24(1974), 424-429. · Zbl 0308.52006
[8] B. Grünbaum and G. C. Shephard, Patterns on the 2-sphere.Mathematika 28(1981), 1-35. · doi:10.1112/S0025579300015321
[9] E. Jucovi?, Self-dual K-polyhedra. [In Russian, with summary in German]Matematicko-fyzikálny ?asopis 12(1962), 1-22.
[10] E. Jucovi?, O mnohostenoch bez opisanej gulovej plochy. [On non-inscribable polyhedra.]Matematicko-fyzikálny ?asopis 15(1965), 90-94.
[11] E. Jucovi?, O mnogostenoch bez opisanej gulovej plochy. II. [On non-inscribable polyhedra. II.]Matematicko-fyzikálny ?asopis 16(1966), 229-234.
[12] E. Jucovi?, Bemerkung zu einem Satz von Steinitz.Elemente der Math. 22(1967), 39.
[13] E. Jucovi?, On polyhedral surfaces which are not inscribable in spherical shells. Problèmes Combinatoires et Théorie des Graphes, Colloques Internat. C. N. R. S. No. 260. Paris 1978, pp. 251-253.
[14] E. Jucovi?, Konvexne mnohosteny. [Convex polyhedra.] Slovenska Akademia Vied, Bratislava 1981. · Zbl 0468.52007
[15] P. McMullen and G. C. Shephard, Convex Polytopes and the Upper Bound Conjecture. London Math. Soc. Lecture Note Series Vol.3, Cambridge Univ. Press 1971. · Zbl 0217.46702
[16] W. Moser, Research Problems in Discrete Geometry. Mimeographed notes, McGill University, 1981.
[17] E. Schulte, Higher-dimensional analogues of Steinitz’s theorem about non-inscribable polytopes. (To appear). · Zbl 0631.52006
[18] S. ?evec, On the non-inscribability of certain families of polyhedra. [In Russian]Math. Slovaca 32(1982), 23-34.
[19] G. C. Shephard, Twenty problems on convex polyhedra.Math Gazette 52(1968), 136-147 and 359-367. · Zbl 0161.41604 · doi:10.2307/3612678
[20] J. Steiner, Systematische Entwickelung der Abhängigkeit geometrischer Gestalten von einander. Fincke, Berlin 1832. (= Gesammelte Werke, Vol.1, Reimer, Berlin 1881, pp. 229-458; see, in particular, p.454).
[21] E. Steinitz, Über isoperimetrische Probleme bei konvexen Polyedern.J. reine angew. Math. 158(1927), 129-153 and 159(1928), 133-143. · JFM 53.0480.02 · doi:10.1515/crll.1927.158.129
[22] T. Tarnai, Spherical grids of triangular network.Acta Techn. Acad. Sci. Hungar. 76(1974), 307-336. · Zbl 0312.52016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.