A structural property of convex 3-polytopes (original) (raw)
Convex 3-polytopes with exactly two types of edges
Discrete Mathematics, 1990
We consider convex 3-polytopes with exactly two types of edges. The questions of the existence of such 3-polytopes are solved. The cardinalities of all classes are determined. 0012-365X/90/$03.50 0 1990 -Elsevier Science Publishers B.V. (North-Holland)
On the simplicial 3-polytopes with only two types of edges
Discrete Mathematics, 1984
For some families of graphs of simplicial 3-polytopes with two types of edges structural properties are described, for other ones their cardinality is determined. Griinbaum and Motzkin [3], Griinbaum and Zaks [4], and Malkevitch [6] investigated the structural properties of trivalent planar graphs with at most two types of faces. It seems that the knowledge of the structure of such graphs is useful for other reasons as well (cf., e.g. Griinbaum [1, 2], Jucovi~ [5], Owens [7], Zaks [8]). The dual problem may be formulated as follows: Characterize simplicial planar graphs with at most two types of vertices. (A planar graph is simplicial if all its faces are triangles.)
A Continuous d-Step Conjecture for Polytopes
Discrete & Computational Geometry, 2009
The curvature of a polytope, defined as the largest possible total curvature of the associated central path, can be regarded as the continuous analogue of its diameter. We prove the analogue of the result of Klee and Walkup. Namely, we show that if the order of the curvature is less than the dimension d for all polytope defined by 2d inequalities and for all d, then the order of the curvature is less that the number of inequalities for all polytopes.
A generalized lower‐bound conjecture for simplicial polytopes
Mathematika, 1971
Let P be a simplicial d-polytope, and, for – 1 ≤ j < d, let fj(P) denote the number of j-faces of P (with f_1 (P) = 1). For k = 0, ..., [½d] – 1, we defineand conjecture thatgk(d + 1)(P) ≥ 0,with equality in the k-th relation if and only if P can be subdivided into a simplicial complex, all of whose simplices of dimension at most d – k – 1 are faces of P. This conjecture is compared with the usual lower-bound conjecture, evidence in support of the conjecture is given, and it is proved that any linear inequality satisfied by the numbers fj(P) is a consequence of the linear inequalities given above.
Polytopes and arrangements: Diameter and curvature
Operations Research Letters, 2008
We introduce a continuous analogue of the Hirsch conjecture and a discrete analogue of the result of Dedieu, Malajovich and Shub. We prove a continuous analogue of the result of Holt and Klee, namely, we construct a family of polytopes which attain the conjectured order of the largest total curvature.
2002
The concept of perfection of a polytope was introduced by S. A. Robertson. Intuitively speaking, a polytope P is perfect if and only if it cannot be deformed to a polytope of different shape without changing the action of its symmetry group G(P ) on its face-lattice F (P ). By Rostami's conjecture, the perfect 4-polytopes form a particular set of Wythoffian polytopes. In the present paper first this known set is briefly surveyed. In the rest of the paper two new classes of perfect 4-polytopes are constructed and discussed, hence Rostami's conjecture is disproved. It is emphasized that in contrast to an existing opinion in the literature, the classification of perfect 4-polytopes is not complete as yet.