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Inventiones mathematicae

, Volume 113, Issue 1, pp 419–444 | Cite as

On simple polytopes

  • Peter McMullen
Article

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 spacesΞr(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 ofΞ1(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.

Keywords

Euclidean Space Striking Feature Mixed Volume Quadratic Inequality Simple Polytopes 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Peter McMullen
    • 1
  1. 1.Department of MathematicsUniversity College LondonLondonUK

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