Determination of the element numbers of the regular polytopes (original) (raw)
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Monatshefte f�r Mathematik, 1990
We call a convex subset N of a convex d-polytope P c E d a k-nucleus of P if N meets every k-face of P, where 0 < k < d. We note that P has disjoint k-nuclei if and only if there exists a hyperplane in E d which bisects the (relative) interior of every k-face of P, and that this is possible only if/~-/~< k ~< d-1.
Construction theorems for polytopes
Israel Journal of Mathematics, 1984
Certain construction theorems are represented, which facilitate an inductive combinatorial construction of polytopes. That is, applying the constructions to a d-polytope with n vertices, given combinatorially, one gets many combinatorial d-polytopes-and polytopes only-with n + I vertices. The constructions are strong enough to yield from the 4-simplex all the 1330 4-polytopes with up to 8 vertices.
On 2-Level Polytopes Arising in Combinatorial Settings
SIAM Journal on Discrete Mathematics
2-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some 2-level polytopes arising in combinatorial settings. Our first contribution is proving that f0(P)f d−1 (P) ≤ d2 d+1 for a large collection of families of such polytopes P. Here f0(P) (resp. f d−1 (P)) is the number of vertices (resp. facets) of P , and d is its dimension. Whether this holds for all 2-level polytopes was asked in [7], and experimental results from [16] showed it true for d ≤ 7. The key to most of our proofs is a deeper understanding of the relations among those polytopes and their underlying combinatorial structures. This leads to a number of results that we believe to be of independent interest: a trade-off formula for the number of cliques and stable sets in a graph; a description of stable matching polytopes as affine projections of certain order polytopes; and a linear-size description of the base polytope of matroids that are 2-level in terms of cuts of an associated tree.
Polytopes with prescribed contents of (n−1)-facets
Israel Journal of Mathematics, 1978
If an n-dimensional polytope has facets of area A1,A2," ",A., then 2A~ < A, + ---+ A. for i = 1, 9 -., m. We show here that conversely these inequalities also ensure the existence of a polytope having these areas.
On Vertices and Facets of Combinatorial 2-Level Polytopes
Lecture Notes in Computer Science, 2016
2-level polytopes naturally appear in several areas of mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. We investigate upper bounds on the product of the number of facets f d−1 (P) and the number of vertices f0(P), where d is the dimension of a 2-level polytope P. This question was first posed in [3], where experimental results showed f0(P)f d−1 (P) ≤ d2 d+1 up to d = 6. We show that this bound holds for all known (to the best of our knowledge) 2-level polytopes coming from combinatorial settings, including stable set polytopes of perfect graphs and all 2-level base polytopes of matroids. For the latter family, we also give a simple description of the facet-defining inequalities. These results are achieved by an investigation of related combinatorial objects, that could be of independent interest.