论文标题
强烈常规CW球的下边界定理，最多$ 2D+1 $顶点
A Lower Bound Theorem for strongly regular CW spheres with up to $2d+1$ vertices
论文作者
论文摘要
1967年，格伦鲍姆（Grünmbaum \ [ϕ_K（D+S，D）= {D+1 \选择K+1}+{D \选择K+1} - {D+1-S \选择K+1} \] $ K $ -Faces。作者最近证明了这种猜想以及平等案例的表征。在本文中，建立了该结果的几个扩展。具体而言，证明具有钻石属性的格子（例如，抽象多型）和$ d+s \ leq 2d $原子具有至少$ ϕ_k（d+s，d）$等级$ k+1 $的元素。此外，在强烈规则的CW复合物的面部晶格的情况下，代表正常的伪雄夫，最高$ 2D $顶点，给出了平等案例的表征。最后，获得了$ k $的$ k $ - 代表普通的伪曼if夫的$ k $ face的尖锐界限，并获得了$ 2D+1 $顶点的正常伪行。这些边界由某些多型的面部数字给出，并带有$ 2D+1 $顶点。
In 1967, Grünmbaum conjectured that any $d$-dimensional polytope with $d+s\leq 2d$ vertices has at least \[ϕ_k(d+s,d) = {d+1 \choose k+1 }+{d \choose k+1 }-{d+1-s \choose k+1 } \] $k$-faces. This conjecture along with the characterization of equality cases was recently proved by the author. In this paper, several extensions of this result are established. Specifically, it is proved that lattices with the diamond property (for example, abstract polytopes) and $d+s\leq 2d$ atoms have at least $ϕ_k(d+s,d)$ elements of rank $k+1$. Furthermore, in the case of face lattices of strongly regular CW complexes representing normal pseudomanifolds with up to $2d$ vertices, a characterization of equality cases is given. Finally, sharp lower bounds on the number of $k$-faces of strongly regular CW complexes representing normal pseudomanifolds with $2d+1$ vertices are obtained. These bounds are given by the face numbers of certain polytopes with $2d+1$ vertices.
