Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

4 Issues per year

IMPACT FACTOR 2017: 0.734

CiteScore 2017: 0.70

SCImago Journal Rank (SJR) 2017: 0.695
Source Normalized Impact per Paper (SNIP) 2017: 0.891

Mathematical Citation Quotient (MCQ) 2017: 0.62

See all formats and pricing
More options …
Ahead of print


Additive structures on f-vector sets of polytopes

Günter M. Ziegler
Published Online: 2018-10-25 | DOI: https://doi.org/10.1515/advgeom-2018-0025


We show that the f-vector sets of d-polytopes have non-trivial additive structure: They span affine lattices and are embedded in monoids that we describe explicitly. Moreover, for many large subclasses, such as the simple polytopes, or the simplicial polytopes, there are monoid structures on the set of f-vectors by themselves: “addition of f-vectors minus the f-vector of the d-simplex” always yields a new f-vector. For general 4-polytopes, we show that the modified addition operation does not always produce an f-vector, but that the result is always close to an f-vector. In this sense, the set of f-vectors of all 4-polytopes forms an “approximate affine semigroup”. The proof relies on the fact for d = 4 every d-polytope, or its dual, has a “small facet”. This fails for d > 4.

We also describe a two further modified addition operations on f-vectors that can be geometrically realized by glueing corresponding polytopes. The second one of these may yield a semigroup structure on the f-vector set of all 4-polytopes.

Keywords: Polytopes; face numbers

MSC 2010: 52A25


  • [1]

    A. Altshuler, L. Steinberg, Enumeration of the quasisimplicial 3-spheres and 4-polytopes with eight vertices. Pacific J. Math. 113 (1984), 269–288. MR749536 Zbl 0512.52004Google Scholar

  • [2]

    A. Altshuler, L. Steinberg, The complete enumeration of the 4-polytopes and 3-spheres with eight vertices. Pacific J. Math. 117 (1985), 1–16. MR777434 Zbl 0512.52003Google Scholar

  • [3]

    E. K. Babson, C. S. Chan, Counting faces of cubical spheres modulo two. Discrete Math. 212 (2000), 169–183. MR1748648 Zbl 0947.52004Google Scholar

  • [4]

    D. Barnette, The projection of the f-vectors of 4-polytopes onto the(E, S)-plane. Discrete Math. 10 (1974), 201–216. MR0353148 Zbl 0294.52008Google Scholar

  • [5]

    D. Barnette, J. R. Reay, Projections of f-vectors of four-polytopes. J. Combin. Theory Ser. A 15 (1973), 200–209. MR0320890 Zbl 0263.05030Google Scholar

  • [6]

    M. M. Bayer, L. J. Billera, Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets. Invent. Math. 79 (1985), 143–157. MR774533 Zbl 0543.52007Google Scholar

  • [7]

    M. M. Bayer, C. W. Lee, Combinatorial aspects of convex polytopes. In: Handbook of convex geometry, Vol. A, B, 485–534, North-Holland 1993. MR1242988 Zbl 0789.52014Google Scholar

  • [8]

    A. Björner, Face numbers of complexes and polytopes. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 1408–1418, Amer. Math. Soc. 1987. MR934345 Zbl 0672.52005Google Scholar

  • [9]

    A. Björner, Partial unimodality for f-vectors of simplicial polytopes and spheres. In: Jerusalem combinatorics ‘93, volume 178 of Contemp. Math., 45–54, Amer. Math. Soc. 1994. MR1310573 Zbl 0815.52011Google Scholar

  • [10]

    A. Björner, S. Linusson, The number of k-faces of a simple d-polytope. Discrete Comput. Geom. 21 (1999), 1–16. MR1661283 Zbl 0935.52010Google Scholar

  • [11]

    G. Blind, R. Blind, Gaps in the numbers of vertices of cubical polytopes. I. Discrete Comput. Geom. 11 (1994), 351–356. MR1271640 Zbl 0807.52009Google Scholar

  • [12]

    P. Brinkmann, G. M. Ziegler, Small f-vectors of 3-spheres and of 4-polytopes. Math. Comp. 87 (2018), 2955–2975. MR3834694 Zbl 06912364Google Scholar

  • [13]

    W. Bruns, J. Gubeladze, N. V. Trung, Problems and algorithms for affine semigroups. Semigroup Forum 64 (2002), 180–212. MR1876854 Zbl 1018.20048Google Scholar

  • [14]

    J. Eckhoff, Combinatorial properties of f-vectors of convex polytopes. Unpublished manuscript, Dortmund 1985.Google Scholar

  • [15]

    J. Eckhoff, Combinatorial properties of f-vectors of convex polytopes. Normat 54 (2006), 146–159. MR2288936Google Scholar

  • [16]

    B. Grünbaum, Convex polytopes. Springer 2003. MR1976856 Zbl 1024.52001Google Scholar

  • [17]

    M. Henk, J. Richter-Gebert, G. M. Ziegler, Basic properties of convex polytopes. In: Handbook of discrete and computational geometry, 383–413, CRC, Boca Raton, FL 2017. MR1730169 Zbl 0911.52007Google Scholar

  • [18]

    W. Höhn, Winkel und Winkelsumme im n-dimensionalen euklidischen Simplex. Thesis, Eidgenössische Technische Hochschule Zürich 1953. MR0056299Google Scholar

  • [19]

    A. Höppner, G. M. Ziegler, A census of flag-vectors of 4-polytopes. In: Polytopes—combinatorics and computation (Oberwolfach, 1997), volume 29 of DMV Sem., 105–110, Birkhäuser 2000. MR1785294 Zbl 0966.52008Google Scholar

  • [20]

    G. Kalai, Rigidity and the lower bound theorem. I. Invent. Math. 88 (1987), 125–151. MR877009 Zbl 0624.52004Google Scholar

  • [21]

    V. Klee, A combinatorial analogue of Poincaré’s duality theorem. Canad. J. Math. 16 (1964), 517–531. MR0189039 Zbl 0134.42403Google Scholar

  • [22]

    T. Kövari, V. T. Sós, P. Turán, On a problem of K. Zarankiewicz. Colloquium Math. 3 (1954), 50–57. MR0065617 Zbl 0055.00704Google Scholar

  • [23]

    E. Miller, B. Sturmfels, Combinatorial commutative algebra. Springer 2005. MR2110098 Zbl 1090.13001Google Scholar

  • [24]

    H. Sjöberg, G. M. Ziegler, Semi-algebraic sets of f-vectors. Preprint, 10 pages, August 2017. arXiv:1711.01864 [math.MG]Google Scholar

  • [25]

    E. Steinitz, Über die Eulerschen Polyederrelationen. Archiv der Mathematik und Physik 11 (1906), 86–88. JFM 37.0500.01Google Scholar

  • [26]

    A. Werner, Linear Constraints on Face Numbers of Polytopes. PhD thesis, TU Berlin, 2009. iv+185 pages, http://opus.kobv.de/tuberlin/volltexte/2009/2263/Google Scholar

  • [27]

    G. M. Ziegler, Lectures on polytopes. Springer 1995. MR1311028 Zbl 0823.52002Google Scholar

  • [28]

    G. M. Ziegler, Face numbers of 4-polytopes and 3-spheres. In: Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), 625–634, Higher Ed. Press, Beijing 2002. MR1957565 Zbl 1006.52006Google Scholar

About the article

Received: 2017-09-13

Revised: 2018-07-01

Published Online: 2018-10-25

Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0025.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in