Summary
LetP be a simpled-polytope ind-dimensional euclidean space\(\mathbb{E}^d \), and let Π(P) be the subalgebra of the polytope algebra Π generated by the classes of summands ofP. It is shown that the dimensions of the weight spacesΞ r(P) of Π(P) are theh-numbers ofP, which describe the Dehn-Sommerville equations between the numbers of faces ofP, and reflect the duality betweenΞ r (P) andΞ d-r (P). Moreover, Π(P) admits a Lefschetz decomposition under multiplication by the element ofΞ 1(P) corresponding toP itself, which yields a proof of the necessity of McMullen's conditions in theg-theorem on thef-vectors of simple polytopes. The Lefschetz decomposition is closely connected with the new Hodge-Riemann-Minkowski quadratic inequalities between mixed volumes, which generalize Minkowski's second inequality; also proved are analogous generalizations of the Aleksandrov-Fenchel inequalities. A striking feature is that these are obtained without using Brunn-Minkowski theory; indeed, the Brunn-Minkowski theorem (without characterization of the cases of equality) can be deduced from them. The connexion found between Π(P) and the face ring of the dual simplicial polytopeP * enables this ring to be looked at in two ways, and a conjectured formulation of theg-theorem in terms of a Gale diagram ofP * is also established.
Similar content being viewed by others
References
Aleksandrov, A.D.: Zur Theorie der gemischten Volumina von konvexen Körpern, II: Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen (in Russian) Mat. Sb. Nov. Ser.2, 1205–1238 (1937)
Barnette, D.W.: A proof of the lower bound conjecture for convex polytopes. Pac. J. Math.46, 349–354 (1973)
Billera, L.J., Lee, C.W.: A proof of the sufficiency of McMullen's conditions forf-vectors of simplicial convex polytopes. J. Comb. Theory, Ser. A31, 237–255 (1981)
Bonnesen, T., Fenchel, W.: Theorie der konvexen Körper. Berlin Heidelberg New York: Springer 1934
Filliman, P.: Rigidity and the Alexandrov-Fenchel inequality. Monatsh. Math.113, 1–22 (1992)
Griffiths, P., Harris, J.: Principles of Algebraic Geometry. New York: Wiley-Interscience 1978
Grünbaum, B.: Convex Polytopes. New York: Wiley-Interscience 1967
Kalai, G.: Rigidity and the lower bound theorem. I. Invent. Math.88, 125–151 (1987)
Kind, B., Kleinschmidt, P.: Cohen-Macaulay-Komplexe und ihre Parametrisierung. Math. Z167, 173–179 (1979)
Lee, C.W.: Some recent results on convex polytopes. Contemp. Math.114, 3–19 (1990)
Macaulay, F.S.: Some properties of enumeration in the theory of modular systems. Proc. Lond. Math. Soc., I. Ser.26, 531–555 (1927)
McMullen, P.: The maximum numbers of faces of a convex polytope. Mathematika17, 179–184 (1970)
McMullen, P.: The numbers of faces of simplicial polytopes. Isr. J. Math.9, 559–570 (1971)
McMullen, P.: Representations of polytopes and polyhedral sets. Geom. Dedicata2 83–99 (1973)
McMullen, P.: Transforms, diagrams and representations. In: (eds.) Tölke, J., Wills, J.M. Contributions to Geometry, Boston Stuttgart Basel Birkhäuser pp. 92–130 1979
McMullen, P.: The polytope algebra. Adv. Math.78, 76–130 (1989)
McMullen, P.: Separation in the polytope algebra. Beiträge Algebra Geometrie34, 15–30 (1993)
McMullen, P., Walkup, D.W.: A generalized lower bound conjecture for simplicial polytopes. Mathematika18, 264–273 (1971)
Oda, T., Park, H.S.: Linear Gale transforms and Gelfand-Kapranov-Zelevinskij decompositions. Tôhoku J. Math.43, 375–399 (1991)
Stanley, R.P.: The upper-bound conjecture and Cohen-Macaulay rings. Stud. Appl. Math.54, 135–142 (1975)
Stanley, R.P.: Hilbert functions of graded algebras. Adv. Math.28, 57–83 (1978)
Stanley, R.P.: The numbers of faces of a simplicial convex polytope. Adv. Math.35, 236–238 (1980)
Walkup, D.W.: The lower bound conjecture for 3- and 4-manifolds. Acta Math.125, 75–107 (1970)
Walkup, D.W., Wets, R.J.-B.: Lifting projections of convex polyhedra. Pac. J. Math.28, 465–475 (1969)
Author information
Authors and Affiliations
Additional information
Oblatum 1-X-1992
Rights and permissions
About this article
Cite this article
McMullen, P. On simple polytopes. Invent Math 113, 419–444 (1993). https://doi.org/10.1007/BF01244313
Issue Date:
DOI: https://doi.org/10.1007/BF01244313