Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets

Bayer, Margaret M.; Billera, Louis J.

doi:10.1007/bf01388660

Cited by 152 publications

(255 citation statements)

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“…With Euler's equation and the Generalized Dehn-Sommerville equations [5] it is routine to derive the following inequality for the class of cs 4-polytopes.…”

Section: Rigidity With Symmetry and Flag-vector Inequalitiesmentioning

confidence: 99%

On Kalai’s Conjectures Concerning Centrally Symmetric Polytopes

Sanyal

Werner

Ziegler

2008

Discrete Comput Geom

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In 1989 Kalai stated the three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the "3 d -conjecture". It is well-known that the three conjectures hold in dimensions d ≤ 3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conjectures B and C fail in all dimensions d ≥ 5.

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“…With Euler's equation and the Generalized Dehn-Sommerville equations [5] it is routine to derive the following inequality for the class of cs 4-polytopes.…”

Section: Rigidity With Symmetry and Flag-vector Inequalitiesmentioning

confidence: 99%

On Kalai’s Conjectures Concerning Centrally Symmetric Polytopes

Sanyal

Werner

Ziegler

2008

Discrete Comput Geom

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“…The flag vector is an integral vector with 2 d − 1 components; nevertheless, due to a multitude of linear relations, the generalized Dehn-Sommerville relations [4], the set of all flag-vectors has dimension only F d − 1, where F d denotes the d-th Fibonacci number. In particular, for d ≤ 3 there is no additional information contained in the flag vector, while for 4-polytopes the set of f -vectors is 3-dimensional, but the set of flag vectors is 4-dimensional.…”

Section: Flag Vectorsmentioning

confidence: 99%

“…Indeed, F d is not just the set of all integral points in a convex set, since some of the constraints, such as f 1 ≤ f0 2 , are concave rather than convex. Also, some of the 2-dimensional coordinate projections of F 4 show "holes" that cannot be explained by such systematic inequalities; compare Grünbaum [7,Section 10.4], Bayer [3], and Höppner and Ziegler [8]. …”

Section: Flag Vectorsmentioning

confidence: 99%

Projected products of polygons

Ziegler¹

2004

Electron. Res. Announc. Amer. Math. Soc.

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Abstract.It is an open problem to characterize the cone of f -vectors of 4-dimensional convex polytopes. The question whether the "fatness" of the fvector of a 4-polytope can be arbitrarily large is a key problem in this context.Here we construct a 2-parameter family of 4-dimensional polytopes π(P 2r n ) with extreme combinatorial structure. In this family, the "fatness" of the fvector gets arbitrarily close to 9; an analogous invariant of the flag vector, the "complexity," gets arbitrarily close to 16.The polytopes are obtained from suitable deformed products of even polygons by a projection to R 4 .

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“…Their key insight is to replace flag enumeration with Euler flag enumeration, that is, a chain of strata is weighted by the Euler characteristic of each link in the chain; see Theorem 5.4. The classical results for the generalized Dehn-Sommerville relations and the cd-index [1,2,31] carry over to this setting.…”

Section: Introductionmentioning

confidence: 98%