Skip to main content
SpringerLink
Log in

Relative Stanley–Reisner theory and Upper Bound Theorems for Minkowski sums

  • Published:
Publications mathématiques de l'IHÉS Aims and scope Submit manuscript

Abstract

In this paper we settle two long-standing questions regarding the combinatorial complexity of Minkowski sums of polytopes: We give a tight upper bound for the number of faces of a Minkowski sum, including a characterization of the case of equality. We similarly give a (tight) upper bound theorem for mixed facets of Minkowski sums. This has a wide range of applications and generalizes the classical Upper Bound Theorems of McMullen and Stanley.

Our main observation is that within (relative) Stanley–Reisner theory, it is possible to encode topological as well as combinatorial/geometric restrictions in an algebraic setup. We illustrate the technology by providing several simplicial isoperimetric and reverse isoperimetric inequalities in addition to our treatment of Minkowski sums.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. K. A. Adiprasito, Toric chordality, preprint, arXiv:1503.06640.

  2. K. A. Adiprasito and B. Benedetti, Subdivisions, shellability, and collapsibility of products, Combinatorica, February 2012, to appear, available at arXiv:1202.6606.

  3. K. A. Adiprasito, A. Björner and A. Goodarzi, Face numbers of sequentially Cohen–Macaulay complexes and Betti numbers of componentwise linear ideals, preprint, arXiv:1502.01183.

  4. K. A. Adiprasito, P. Brinkmann, A. Padrol and R. Sanyal, Mixed faces and colorful depth, 2015, in preparation.

  5. D. Barnette, P. Kleinschmidt and C. W. Lee, An upper bound theorem for polytope pairs, Math. Oper. Res., 11 (1986), 451–464.

    Article  MathSciNet  MATH  Google Scholar 

  6. L. J. Billera and C. W. Lee, Sufficiency of McMullen’s conditions for \(f\)-vectors of simplicial polytopes, Bull. Am. Math. Soc., New Ser., 2 (1980), 181–185.

    Article  MathSciNet  MATH  Google Scholar 

  7. L. J. Billera and C. W. Lee, The numbers of faces of polytope pairs and unbounded polyhedra, Eur. J. Comb., 2 (1981), 307–322.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Björner, Shellable and Cohen-Macaulay partially ordered sets, Trans. Am. Math. Soc., 260 (1980), 159–183.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Björner, Nerves, fibers and homotopy groups, J. Comb. Theory, Ser. A, 102 (2003), 88–93.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Björner, A comparison theorem for \(f\)-vectors of simplicial polytopes, Pure Appl. Math. Q., 3 (2007), 347–356.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Björner, M. L. Wachs and V. Welker, Poset fiber theorems, Trans. Am. Math. Soc., 357 (2005), 1877–1899.

    Article  MathSciNet  MATH  Google Scholar 

  12. K. Borsuk, On the imbedding of systems of compacta in simplicial complexes, Fundam. Math., 35 (1948), 217–234.

    MathSciNet  MATH  Google Scholar 

  13. H. Bruggesser and P. Mani, Shellable decompositions of cells and spheres, Math. Scand., 29 (1971), 197–205, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  14. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993.

    MATH  Google Scholar 

  15. R. C. Buck, Partition of space, Am. Math. Mon., 50 (1943), 541–544.

    Article  MathSciNet  MATH  Google Scholar 

  16. R. M. Charney and M. W. Davis, Strict hyperbolization, Topology, 34 (1995), 329–350.

    Article  MathSciNet  MATH  Google Scholar 

  17. D. A. Cox, J. B. Little and H. K. Schenck, Toric Varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, 2011.

    MATH  Google Scholar 

  18. M. W. Davis, The Geometry and Topology of Coxeter Groups, London Mathematical Society Monographs Series, vol. 32, Princeton University Press, Princeton, 2008.

    MATH  Google Scholar 

  19. J. A. de Loera, J. Rambau and F. Santos, Triangulations, Algorithms and Computation in Mathematics, vol. 25, Springer, Berlin, 2010, Structures for algorithms and applications.

    MATH  Google Scholar 

  20. A. M. Duval, Algebraic shifting and sequentially Cohen-Macaulay simplicial complexes, Electron. J. Comb., 3 (1996), 14, research paper r21.

    MathSciNet  MATH  Google Scholar 

  21. D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150, Springer, New York, 1995.

    MATH  Google Scholar 

  22. I. Z. Emiris and J. F. Canny, Efficient incremental algorithms for the sparse resultant and the mixed volume, J. Symb. Comput., 20 (1995), 117–149.

    Article  MathSciNet  MATH  Google Scholar 

  23. K. Fukuda and C. Weibel, \(f\)-Vectors of Minkowski additions of convex polytopes, Discrete Comput. Geom., 37 (2007), 503–516.

    Article  MathSciNet  MATH  Google Scholar 

  24. K. Fukuda and C. Weibel, A linear equation for Minkowski sums of polytopes relatively in general position, Eur. J. Comb., 31 (2010), 565–573.

    Article  MathSciNet  MATH  Google Scholar 

  25. W. Fulton and B. Sturmfels, Intersection theory on toric varieties, Topology, 36 (1997), 335–353.

    Article  MathSciNet  MATH  Google Scholar 

  26. R. Geoghegan, Topological Methods in Group Theory, Graduate Texts in Mathematics, vol. 243, Springer, New York, 2008.

    MATH  Google Scholar 

  27. H.-G. Gräbe, Generalized Dehn-Sommerville equations and an upper bound theorem, Beitr. Algebra Geom., 25 (1987), 47–60.

    MathSciNet  MATH  Google Scholar 

  28. P. Gritzmann and B. Sturmfels, Minkowski addition of polytopes: computational complexity and applications to Gröbner bases, SIAM J. Discrete Math., 6 (1993), 246–269.

    Article  MathSciNet  MATH  Google Scholar 

  29. T. Hibi, Quotient algebras of Stanley-Reisner rings and local cohomology, J. Algebra, 140 (1991), 336–343.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes, in Ring Theory, II, Proc. Second Conf., Univ. Oklahoma, Norman, Okla, 1975, Lecture Notes in Pure and Appl. Math., vol. 26, pp. 171–223, Dekker, New York, 1977.

    Google Scholar 

  31. M. Hochster and J. L. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Adv. Math., 13 (1974), 115–175.

    Article  MathSciNet  MATH  Google Scholar 

  32. A. G. Hovanskiĭ, Newton polyhedra, and the genus of complete intersections, Funkc. Anal. Prilozh., 12 (1978), 51–61.

    MathSciNet  Google Scholar 

  33. S. B. Iyengar, G. J. Leuschke, A. Leykin, C. Miller, E. Miller, A. K. Singh and U. Walther, Twenty-Four Hours of Local Cohomology, Graduate Studies in Mathematics, vol. 87, American Mathematical Society, Providence, 2007.

    MATH  Google Scholar 

  34. W. Julian, R. Mines and F. Richman, Alexander duality, Pac. J. Math., 106 (1983), 115–127.

    Article  MathSciNet  MATH  Google Scholar 

  35. G. Kalai, The diameter of graphs of convex polytopes and \(f\)-vector theory, in Applied Geometry and Discrete Mathematics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4, pp. 387–411, Amer. Math. Soc., Providence, 1991.

    Google Scholar 

  36. M. I. Karavelas, C. Konaxis and E. Tzanaki, The maximum number of faces of the Minkowski sum of three convex polytopes, J. Comput. Geom., 6 (2015), 21–74.

    MathSciNet  MATH  Google Scholar 

  37. M. I. Karavelas and E. Tzanaki, The maximum number of faces of the Minkowski sum of two convex polytopes, preprint, (2011), Discrete Comput. Geom., to appear, available at doi:10.1007/s00454-015-9726-6.

  38. M. I. Karavelas and E. Tzanaki, A geometric approach for the upper bound theorem for Minkowski sums of convex polytopes, in 31st International Symposium on Computational Geometry, LIPIcs, Leibniz Int. Proc. Inform., vol. 34, pp. 81–95, Schloss Dagstuhl. Leibniz-Zent. Inform, Wadern, 2015.

    Google Scholar 

  39. E. Katz, Tropical intersection theory from toric varieties, Collect. Math., 63 (2012), 29–44.

    Article  MathSciNet  MATH  Google Scholar 

  40. B. Kind and P. Kleinschmidt, Schälbare Cohen-Macauley-Komplexe und ihre Parametrisierung, Math. Z., 167 (1979), 173–179.

    Article  MathSciNet  MATH  Google Scholar 

  41. V. Klee, On the number of vertices of a convex polytope, Can. J. Math., 16 (1964), 701–720.

    Article  MathSciNet  MATH  Google Scholar 

  42. J-C. Latombe, Robot Motion Planning, vol. 25, Kluwer Academic Publishers, Boston, 1991.

    Book  MATH  Google Scholar 

  43. B. Matschke, J. Pfeifle and V. Pilaud, Prodsimplicial-neighborly polytopes, Discrete Comput. Geom., 46 (2011), 100–131.

    Article  MathSciNet  MATH  Google Scholar 

  44. P. McMullen, The maximum numbers of faces of a convex polytope, Mathematika, 17 (1970), 179–184.

    Article  MathSciNet  MATH  Google Scholar 

  45. P. McMullen, Triangulations of simplicial polytopes, Beitr. Algebra Geom., 45 (2004), 37–46.

    MathSciNet  MATH  Google Scholar 

  46. P. McMullen and D. W. Walkup, A generalized lower-bound conjecture for simplicial polytopes, Mathematika, 18 (1971), 264–273.

    Article  MathSciNet  MATH  Google Scholar 

  47. E. Miller, I. Novik and E. Swartz, Face rings of simplicial complexes with singularities, Math. Ann., 351 (2011), 857–875.

    Article  MathSciNet  MATH  Google Scholar 

  48. E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227, Springer, New York, 2005.

    MATH  Google Scholar 

  49. H. Minkowski, Theorie der konvexen körper, insbesondere Begründung ihres Oberflächenbegriffs, Gesammelte Abh. Hermann Minkowski, 2 (1911), 131–229.

    Google Scholar 

  50. M. Miyazaki, Characterizations of Buchsbaum complexes, Manuscr. Math., 63 (1989), 245–254.

    Article  MathSciNet  MATH  Google Scholar 

  51. T. S. Motzkin, Comonotone curves and polyhedra, Bull. Am. Math. Soc., 63 (1957), 35.

    Article  MathSciNet  Google Scholar 

  52. T. S. Motzkin, H. Raiffa, G. L. Thompson and R. M. Thrall, The double description method, in Contributions to the Theory of Games, Annals of Mathematics Studies, vol. 28, pp. 51–73, Princeton University Press, Princeton, 1953.

    Google Scholar 

  53. I. Novik, Remarks on the upper bound theorem, J. Comb. Theory, Ser. A, 104 (2003), 201–206.

    Article  MathSciNet  MATH  Google Scholar 

  54. I. Novik, On face numbers of manifolds with symmetry, Adv. Math., 192 (2005), 183–208.

    Article  MathSciNet  MATH  Google Scholar 

  55. I. Novik and E. Swartz, Applications of Klee’s Dehn–Sommerville relations, Discrete Comput. Geom., 42 (2009), 261–276.

    Article  MathSciNet  MATH  Google Scholar 

  56. I. Novik and E. Swartz, Face numbers of pseudomanifolds with isolated singularities, Math. Scand., 110 (2012), 198–222.

    Article  MathSciNet  MATH  Google Scholar 

  57. P. Pedersen and B. Sturmfels, Product formulas for resultants and Chow forms, Math. Z., 214 (1993), 377–396.

    Article  MathSciNet  MATH  Google Scholar 

  58. G. A. Reisner, Cohen-Macaulay quotients of polynomial rings, Adv. Math., 21 (1976), 30–49.

    Article  MathSciNet  MATH  Google Scholar 

  59. C. P. Rourke and B. J. Sanderson, Introduction to Piecewise-Linear Topology, Ergebnisse Series, vol. 69, Springer, New York, 1972.

    Book  MATH  Google Scholar 

  60. R. Sanyal, Topological obstructions for vertex numbers of Minkowski sums, J. Comb. Theory, Ser. A, 116 (2009), 168–179.

    Article  MathSciNet  MATH  Google Scholar 

  61. P. Schenzel, On the number of faces of simplicial complexes and the purity of Frobenius, Math. Z., 178 (1981), 125–142.

    Article  MathSciNet  MATH  Google Scholar 

  62. P. Schenzel, Dualisierende Komplexe in der lokalen Algebra und Buchsbaum-Ringe, Lecture Notes in Mathematics, vol. 907, Springer, Berlin, 1982, with an English summary.

    Book  MATH  Google Scholar 

  63. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993.

    Book  MATH  Google Scholar 

  64. R. P. Stanley, The upper bound conjecture and Cohen-Macaulay rings, Stud. Appl. Math., 54 (1975), 135–142.

    Article  MathSciNet  MATH  Google Scholar 

  65. R. P. Stanley, Generalized \(H\)-vectors, intersection cohomology of toric varieties, and related results, in Commutative Algebra and Combinatorics, Adv. Stud. Pure Math., vol. 11, Kyoto, 1985, pp. 187–213, North-Holland, Amsterdam, 1987.

    Google Scholar 

  66. R. P. Stanley, A monotonicity property of \(h\)-vectors and \(h^{*}\)-vectors, Eur. J. Comb., 14 (1993), 251–258.

    Article  MathSciNet  MATH  Google Scholar 

  67. R. P. Stanley, Combinatorics and Commutative Algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Boston, 1996.

    MATH  Google Scholar 

  68. R. Steffens and T. Theobald, Combinatorics and genus of tropical intersections and Ehrhart theory, SIAM J. Discrete Math., 24 (2010), 17–32.

    Article  MathSciNet  MATH  Google Scholar 

  69. J. Steiner, Einige Gesetze über die Theilung der Ebene und des Raumes, J. Reine Angew. Math., 1 (1826), 349–364.

    Article  MathSciNet  Google Scholar 

  70. B. Sturmfels, Solving Systems of Polynomial Equations, CBMS Regional Conference Series in Mathematics, vol. 97, Conference Board of the Mathematical Sciences, Washington, 2002.

    MATH  Google Scholar 

  71. E. Swartz, Lower bounds for \(h\)-vectors of \(k\)-CM, independence, and broken circuit complexes, SIAM J. Discrete Math., 18 (2005), 647–661.

    Article  MathSciNet  MATH  Google Scholar 

  72. C. Weibel, Maximal \(f\)-vectors of Minkowski sums of large numbers of polytopes, Discrete Comput. Geom., 47 (2012), 519–537.

    Article  MathSciNet  MATH  Google Scholar 

  73. E. C. Zeeman, Seminar on Combinatorial Topology, Institut des Hautes Etudes Scientifiques, Paris, 1966.

    MATH  Google Scholar 

  74. G. M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Springer, New York, 1995, Revised edition, 1998; seventh updated printing 2007.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Einstein Institute for Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel

    Karim A. Adiprasito

  2. Fachbereich Mathematik und Informatik, Freie Universität Berlin, Berlin, Germany

    Raman Sanyal

Authors
  1. Karim A. Adiprasito
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Raman Sanyal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karim A. Adiprasito.

Additional information

K. Adiprasito was supported by an EPDI/IPDE postdoctoral fellowship, a Minerva fellowship of the Max Planck Society, the DFG within the research training group “Methods for Discrete Structures” (GRK1408) and by the Romanian NASR, CNCS—UEFISCDI, project PN-II-ID-PCE-2011-3-0533.

R. Sanyal was supported by European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013) / ERC grant agreement no 247029 and by the DFG-Collaborative Research Center, TRR 109 “Discretization in Geometry and Dynamics”.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Adiprasito, K.A., Sanyal, R. Relative Stanley–Reisner theory and Upper Bound Theorems for Minkowski sums. Publ.math.IHES 124, 99–163 (2016). https://doi.org/10.1007/s10240-016-0083-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10240-016-0083-7

Keywords

Search

Navigation