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Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets
| Content Provider | Semantic Scholar |
|---|---|
| Author | Bayer, Margaret M. Billera, Louis J. |
| Copyright Year | 1985 |
| Abstract | This paper generalizes the Dehn-Sommerville equations for simplicial spheres to related classes of objects. The underlying motivation is to understand the combinatorial structure of arbitrary polytopes, that is, polytopes that are not necessarily simplicial. Towards this end we determine the affine span of the extended f-vectors of d-polytopes. A polytope is the convex hull of finitely many points in R e . We will generally consider a polytope of affine dimension d to be a subset of Re; this is referred to as a d-polytope. A face of a polytope is the intersection of a supporting hyperplane with the polytope. For the most part we identify a polytope P with the abstract cell complex (or lattice) realized by the boundary of P, and write a face of P as the set of vertices of P it contains. That is, we will shorten F = c o n v { v 0 . . . . . Vk} to F={Vo,.. . ,Vk} when there is no risk of confusion. By convention, the empty set is considered a ( -1)-dimensional face, and the polytope itself is a d-dimentional face; these faces will be called improper faces of P. A polytope P is called simplicial if each of its faces, except possibly P itself, is a simplex (the convex hull of affinely independent points). We will write Y (respectively, ~ ) for the set of all (respectively, all simplicial) d-polytopes. The number of/-dimensional faces (or/-faces) of a polytope P is written f , and f(P)=(fo,fl . . . . . fd-1) is called the f-vector of P. The set off -vectors of all (simplicial) polytopes is written f ( ~ e ) (f(~d)). A certain transformation on the f-vectors of simplicial polytopes has arisen in a number of different contexts, and will play an important part here. For a d-polytope P define the h-vector h(P)=(ho, hl,...,he) b y h i = i ( 1 ) i s ( d j ) s= o d i f j 1 (here we use the convention |
| Starting Page | 143 |
| Ending Page | 157 |
| Page Count | 15 |
| File Format | PDF HTM / HTML |
| DOI | 10.1007/BF01388660 |
| Volume Number | 79 |
| Alternate Webpage(s) | https://page-one.springer.com/pdf/preview/10.1007/BF01388660 |
| Alternate Webpage(s) | https://doi.org/10.1007/BF01388660 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
