About f-Vectors of Inscribed Simplicial Polytopes

Discrete & Computational Geometry - Tập 55 Số 3 - Trang 497-521 - 2016
Gonska, Bernd1
1Inst. Math. and Comp. Sci., FU Berlin, Berlin, Germany

Tóm tắt

We show that for every simplicial polytope an inscribed simplicial polytope exists that has the same dimension, number of vertices, number of edges, and number of 2-faces. This proves that the g-theorem for simplicial polytopes also holds for the class of inscribed simplicial polytopes (up to dimension 7). The proof includes an incremental construction scheme for Delaunay triangulations.

Tài liệu tham khảo

citation_journal_title=Isr. J. Math.; citation_title=The minimum number of vertices of a simple polytope; citation_author=DW Barnette; citation_volume=10; citation_publication_date=1971; citation_pages=121-125; citation_doi=10.1007/BF02771522; citation_id=CR1 citation_journal_title=Pac. J. Math.; citation_title=A proof of the lower bound conjecture for convex polytopes; citation_author=DW Barnette; citation_volume=46; citation_publication_date=1973; citation_pages=349-354; citation_doi=10.2140/pjm.1973.46.349; citation_id=CR2 citation_title=Geometry I; citation_publication_date=2009; citation_id=CR3; citation_author=M Berger; citation_publisher=Springer citation_title=Geometry II; citation_publication_date=2009; citation_id=CR4; citation_author=M Berger; citation_publisher=Springer citation_journal_title=Bull. Am. Math. Soc. (N.S.); citation_title=Sufficiency of McMullen’s conditions for -vectors of simplicial polytopes; citation_author=LJ Billera, CW Lee; citation_volume=2; citation_publication_date=1980; citation_pages=181-185; citation_doi=10.1090/S0273-0979-1980-14712-6; citation_id=CR5 citation_journal_title=J. Comb. Theory, Ser. A; citation_title=A proof of the sufficiency of McMullen’s conditions for -vectors of simplicial convex polytopes; citation_author=LJ Billera, CW Lee; citation_volume=31; citation_publication_date=1981; citation_pages=237-255; citation_doi=10.1016/0097-3165(81)90058-3; citation_id=CR6 citation_journal_title=Inf. Process. Lett.; citation_title=Voronoi diagrams from convex hulls; citation_author=KQ Brown; citation_volume=9; citation_publication_date=1979; citation_pages=223-228; citation_doi=10.1016/0020-0190(79)90074-7; citation_id=CR7 citation_title=Triangulations; citation_publication_date=2010; citation_id=CR8; citation_author=JA Loera; citation_author=J Rambau; citation_author=FP Santos; citation_publisher=Springer citation_journal_title=Int. J. Comput. Geom. Appl.; citation_title=A linear-time algorithm for testing the inscribability of trivalent polyhedra; citation_author=MB Dillencourt, WD Smith; citation_volume=5; citation_publication_date=1995; citation_pages=21-36; citation_doi=10.1142/S0218195995000039; citation_id=CR9 citation_journal_title=Discrete Math.; citation_title=Graph-theoretical conditions for inscribability and Delaunay realizability; citation_author=MB Dillencourt, WD Smith; citation_volume=161; citation_publication_date=1996; citation_pages=63-77; citation_doi=10.1016/0012-365X(95)00276-3; citation_id=CR10 citation_title=Geometry and Topology for Mesh Generation; citation_publication_date=2001; citation_id=CR11; citation_author=H Edelsbrunner; citation_publisher=Cambridge University Press Gale, D.: Neighborly and cyclic polytopes. In: Proceedings of Symposia in Pure Mathematics, Vol. VII, pp. 225–232 (1963) Gonska, B., Ziegler, G.M.: Inscribable stacked polytopes. Adv. Geom. 13, 723–740 (2013) citation_title=Convex Polytopes; citation_publication_date=2003; citation_id=CR14; citation_author=B Grünbaum; citation_publisher=Springer citation_journal_title=Bull. Am. Math. Soc. (N.S.); citation_title=A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere; citation_author=CD Hodgson, I Rivin, WD Smith; citation_volume=27; citation_publication_date=1992; citation_pages=246-251; citation_doi=10.1090/S0273-0979-1992-00303-8; citation_id=CR15 citation_journal_title=Mathematika; citation_title=The maximum numbers of faces of a convex polytope; citation_author=P McMullen; citation_volume=17; citation_publication_date=1970; citation_pages=179-184; citation_doi=10.1112/S0025579300002850; citation_id=CR16 citation_journal_title=Topology; citation_title=On geometry of convex ideal polyhedra in hyperbolic 3-space; citation_author=I Rivin; citation_volume=32; citation_publication_date=1993; citation_pages=87-92; citation_doi=10.1016/0040-9383(93)90039-X; citation_id=CR17 citation_journal_title=Ann. Math.; citation_title=A characterization of ideal polyhedra in hyperbolic 3-space; citation_author=I Rivin; citation_volume=143; citation_publication_date=1996; citation_pages=51-70; citation_doi=10.2307/2118652; citation_id=CR18 Rivin, I., Hodgson, C.D.: A characterization of compact convex polyhedra in hyperbolic 3-space/corrigendum. Invent. Math. 111/117, 77–111/359 (1993/1994) Seidel, R.: Exact upper bounds for the number of faces in $$d$$ -dimensional Voronoĭ diagrams. In: Applied Geometry and Discrete Mathematics, vol. 4 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci. American Mathematical Society, Providence, RI, pp. 517–529 (1991) citation_journal_title=Adv. Math.; citation_title=The number of faces of a simplicial convex polytope; citation_author=RP Stanley; citation_volume=35; citation_publication_date=1980; citation_pages=236-238; citation_doi=10.1016/0001-8708(80)90050-X; citation_id=CR21 Steiner, J.: Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander, Gesammelte Werke, Reimer, Berlin (Original publication: Fincke, Berlin (1832)), vol. 1, pp. 229–458 (1881) Steinitz, E.: Über isoperimetrische Probleme bei konvexen Polyedern. J. Reine Angew. Math. 158/159, 129–153/143 (1927/1928) citation_title=Lectures on Polytopes; citation_publication_date=1995; citation_id=CR24; citation_author=GM Ziegler; citation_publisher=Springer Ziegler, G.M.: Face numbers of 4-polytopes and 3-spheres. In: Proceedings of the International Congress of Mathematicians, vol. III, pp. 625–634. Higher Ed. Press, Beijing (2002)