A quasi-polynomial bound for the diameter of graphs of polyhedra

A simple proof of tail--polynomial bounds on the diameter of polyhedra

Let $Δ(d,n)$ denote the maximum diameter of a $d$-dimensional polyhedron with $n$ facets. In this paper, we propose a unified analysis of a recursive inequality about $Δ(d,n)$ established by Kalai and Kleitman in 1992. This yields much simpler proofs of a tail--polynomial and tail--almost--linear bounds on $Δ(d,n)$ which are recently discussed by Gallagher and Kim.

通过Reuleaux多面体研究Borsuk问题与Vázsonyi问题

Borsuk and Vázsonyi problems through Reuleaux polyhedra

本文利用Reuleaux多面体建立了三维空间中Borsuk数为4的临界集与Vázsonyi问题极小结构之间的等价关系，并由此完整刻画了所有Borsuk数为4的有限集合。

Quasi--Euclidean classification of Alcoved Convex Polyhedra

We give the quasi--Euclidean classification of the maximal (with respect to the $f$--vector) alcoved polyhedra. The $f$--vector of these maximal convex bodies is $(20,30,12)$, so they are simple dodecahedra. We find eight quasi--Euclidean classes. This classification, which preserves angles, is finer than the known combinatorial classification (found in 2012 by Jiménez and de la Puente), which has only six classes. Each alcoved polyhedron $\mathcal{P}$ is represented by a unique visualized idempotent matrix $A$. Some 2--minors of $A$ are invariants of $\mathcal{P}$: they are the tropical edge-…

A note on the diameter of convex polytope

This short note extends a recent result (Bonifas et al, On sub-determinants and the diameter of polyhedra, Discrete Computational Geometry, 52, 2014) of an upper bound of the diameter of a convex polytope defined by an integer matrix to a similar upper bound of the diameter of a convex polytope defined by a real matrix. It also shows, by an example, that the new bound may be better than the ones of Bonifas et al.

拟对称网：Kokotsakis多面体等模椭圆型的一种构造性方法

Quasi-symmetric nets: A constructive approach to the equimodular elliptic type of Kokotsakis polyhedra

本文提出拟对称网（QS-nets）来构造四边底等模椭圆型柔性Kokotsakis多面体，给出显式代数刻画和闭式变形参数化，并开发数值管道验证实例。

On origami-like quasi-mechanisms with an antiprismatic skeleton

We study snapping and shaky polyhedra which consist of antiprismatic skeletons covered by polyhedral belts composed of triangular faces only. In detail, we generalize Wunderlich's trisymmetric sandglass polyhedron in analogy to the generalizsation of the Jessen orthogonal icosahedron to Milka's extreme birosette structures, with the additional feature that the belt is developable into the plane as the Kresling pattern. Within the resulting 2-dimensional family of origami-like sandglasses we study the 1-parametric sets of quasi-mechanisms which are either shaky or have an extremal snap, i.e. on…