On low-dimensional faces that high-dimensional polytopes must have
Tóm tắt
We prove that every five-dimensional polytope has a two-dimensional face which is a triangle or a quadrilateral. We state and discuss the following conjecture: For every integerk≥1 there is an integer f(k) such that everyd-polytope,d≥f(k), has ak-dimensional face which is either a simplex or combinatorially isomorphic to thek-dimensional cube. We give some related results concerning facet-forming polytopes and tilings. For example, sharpening a result of Schulte [25] we prove that there is no face to face tiling of ℝ5 with crosspolytopes.
Tài liệu tham khảo
M. Bayer: The extendedf-vector for 4-polytopes,J. Combin. Th. Ser. A,44 (1987), 141–151.
D. Barnette: A simple 4-dimensional nonfacet,Israel J. Math.,7 (1969), 16–20.
D. Barnette: A proof of the lower bound conjecture for convex polytopes,Pacific J. Math.,46 (1973), 349–254.
D. Barnette: Nonfacets for shellable spheres,Israel J. Math.,35 (1980), 286–288.
M. Bayer, andL. Billera: Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets,Invent. Math.,79 (1985), 143–157.
L. Billera, andK. Magurn: Balanced subdivisions and enumeration in balanced spheres,Discrete and Computational Geometry,2 (1987), 297–317.
G. Blind, andR. Blind: Über die minimalan Seitenzahlen von Polytopen ohne dreieckige 2-Seiten,Monastsh. Math.,98 (1984), 179–184.
A. Brøndsted:An Introduction to Convex Polytopes, Graduate text in Mathematics no. 90, Springer-Verlag, New York,1980
V. Chvatal: On certain polytopes associated with graphs,J. Combin. Th. Ser. B,18 (1975), 138–154.
H. S. M. Coxeter:Regular Polytopes, Macmillan, London and New York,1948.
L.Danzer: private communications.
P. J. Federico:Descartes on Polyhedra, Springer-Verlag, New York,1982.
S. Hansen: A Generalization of a theorem of Sylvester on the lines determined by a finite point set,Math. Scan.,16 (1965), 175–180.
M.Gromov: Hyperbolic groups, inEssays in Group Theory, S. Gersten (ed.) 75–264, Springer Verlag, 1987.
B. Grünbaum:Convex Polytopes, Wiley, Interscience, New York,1967.
G. Kalai: Rigidity and the lower bound theorem 1,Invent. Math.,88 (1987), 125–151.
G. Kalai: A new basis of polytopes,J. Combin. Th. Ser. A.,49 (1988), 191–209.
Y. Kupitz: An open problem,Ann. Discr. Math.,20 (1984), 338.
C. Lee: The associahedron and triangulations ofn-gons,Europ. J. Combin.,10 (1989), 551–560.
P. McMullen: On Zonotopes,Trans Amer. Math. Soc.,159 (1971), 559–570.
P. McMullen, andG. C. Shephard:Convex Polytopes and the Upper Bound Theorem, Cambridge Univ. Press, London,1971.
M. A. Perles, G. C. Shephard: Facets and nonfacets of convex polytopes,Acta Math.,119 (1967), 113–145.
E. Schulte: Nontiles and nonfacets for the Euclidean space, spherical complexes and convex polytopes,J. reine angew. Math.,352 (1984), 161–183.
E. Schulte: The existence of nontiles and nonfacets in three dimensions,J. Combin. Th. Ser. A.,38 (1985), 75–81.
R. Stanley: Balanced Cohen-Macaulay complexes,Trans. Amer. Math. Soc.,249 (1979), 139–157.
R. Stanley: Generalizedh-vectors, intersection homology of toric varieties, and related results,Advanced Studies in Pure Mathematics 11, M. Nagata and H. Matsumura (eds.), 187–214, Kinukuniya Company Ltd., Tokyo,1987.
V. V. Nikulin: Classification of arithmetic groups generated by reflections in Lobachevskii spaces,Izv. Akad. Nauk SSSR, Ser. Mat.,45 (1981), 113–142.
V. V. Nikulin: Discrete reflection groups in Lobachevskii spaces and algebraic surfaces,Proc. Int. Cong. of Math., Berkeley,1986 Vol I. 654–671.
A. G. Khovanskii: Hyperplane sections of polyhedra, toroidal manifolds, and discrete groups in Lobachevskii space,Funktsionalnyi Analiz i Ego Prilozheniya,20 (1986), 50–61.
G.Blind, and R.Blind: Convex polytopes without triangular faces,Israel J. Math., to appear.