New results on lower bounds for the number of (at most k)-facets
Oswin Aichholzer, Jesús García, David Orden, Pedro Ramos
备注：
11 pages, 5 figures. Submitted to European Journal of Combinatorics. Changes: last result in the abstract generalized to $d$ dimensions, Section 4 simplified
发表信息：
European Journal of Combinatorics 30:7 (2009), 1568-1574
In this paper we present three different results dealing with the number of $(\leq k)$-facets of a set of points:
1. We give structural properties of sets in the plane that achieve the optimal lower bound $3\binom{k+2}{2}$ of $(\leq k)$-edges for a fixed $0\leq k\leq \lfloor n/3 \rfloor -1$;
2. We give a simple construction showing that the lower bound $3\binom{k+2}{2}+3\binom{k-\lfloor \frac{n}{3} \rfloor+2}{2}$ for the number of $(\leq k)$-edges of a planar point set appeared in [Aichholzer et al. New lower bounds for the number of ($\leq k$)-edges and the rectilinear crossing number of $K_n$. {\em Disc. Comput. Geom.} 38:1 (2007), 1--14] is optimal in the range $\lfloor n/3 \rfloor \leq k \leq \lfloor 5n/12 \rfloor -1$;
3. We show that for $k < \lfloor n/(d+1) \rfloor$ the number of $(\leq k)$-facets of a set of $n$ points in general position in $\mathbb{R}^d$ is at least $(d+1)\binom{k+d}{d}$, and that this bound is tight in the given range of $k$.
We have found the minimal difference $Δ(k) = \min\limits_P (f_{d-1}(P) - f_{0}(P))$ between the number of facets and the number of vertices of a $k$-neighborly $d$-polytope $P$ for the case $f_{0}(P) = d+3$: $Δ(2) = 4$, $Δ(3) = 15$, and $Δ(k) = 2 (k^2 - 1)$ for $k \ge 4$.
We give a complete enumeration of all 2-neighborly $d$-polytopes with $d+9$ and less facets. All of them are realized as 0/1-polytopes, except a 6-polytope $P_{6,10,15}$ with 10 vertices and 15 facets, and pyramids over $P_{6,10,15}$. In particular, we update the lower bounds for the number of facets of a 2-neighborly $d$-polytope $P$ and showed that the number of facets of $P$ is not less than the number of its vertices $f_0(P)$ for $f_0(P) \le d + 10$.
We show that the edge graph of a 6-dimensional polytope with 12 facets has diameter at most 6, thus verifying the d-step conjecture of Klee and Walkup in the case of d=6. This implies that for all pairs (d,n) with n-d \leq 6 the diameter of the edge graph of a d-polytope with n facets is bounded by 6, which proves the Hirsch conjecture for all n-d \leq 6. We show this result by showing this bound for a more general structure -- so-called matroid polytopes -- by reduction to a small number of satisfiability problems.
In this note we prove that the number of combinatorial types of $d$-polytopes with $d+1+α$ vertices and $d+1+β$ facets is bounded by a constant independent of $d$.
In this paper we present a bijection $ω_n$ between two well known families of Catalan objects: the set of facets of the $m$-generalized cluster complex $Δ^m(A_n)$ and the set of dominant regions in the $m$-Catalan arrangement ${\rm Cat}^m(A_n)$, where $m\in\mathbb{N}_{>0}$. In particular, $ω_n$ bijects the facets containing the negative simple root $-α$ to dominant regions having the hyperplane $\{v\in V\mid<v,α>=m\}$ as separating wall. As a result, $ω_n$ restricts to a bijection between the set of facets of the positive part of $Δ^m(A_n)$ and the set of bounded dominant regions in ${\rm Cat}…
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Lattice coverings by parallelotopes containing cube vertices
Institute of Mathematics and Informatics Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Faculty of Information Sciences State University of Library Studies and Information Technologies（图书馆研究与信息技术州立大学信息科学学院）
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