On braxtopes, a class of generalized simplices

根多项形的细分与广义热带定向矩阵oids（扩展摘要）

Subdivisions of root polytopes, generalized tropical oriented matroids, and generalized matching ensembles

本文研究了根多项形的细分与广义热带定向矩阵oids的对应关系，揭示了其一一映射的数学结构。

广义置换多面体的多面体代数

The polytope algebra of generalized permutahedra

本研究探讨广义置换多面体的多面体代数构造，揭示B型广义置换多面体生成族需至少2^(d-1)个满维多面体并找到达此最小值的单纯形生成族，还证明相关关系与Hopf幺半群结构兼容。

On some generalization of the Möbius configuration

The Möbius $(8_4)$ configuration is generalized in a purely combinatorial approach. We consider $(2n_n)$ configurations ${\goth M}_{(n,φ)}$ depending on a permutation $φ$ in the symmetric group $S_n$. Classes of non-isomorphic configurations of this type are determined. The parametric characterization of ${\goth M}_{(n,φ)}$ is given. The uniqueness of the decomposition of ${\goth M}_{(n,φ)}$ into two mutually inscribed $n$-simplices is discussed. The automorphisms of ${\goth M}_{(n,φ)}$ are characterized for $n\geq 3$.

Trianguloids and Triangulations of Root Polytopes

Triangulations of a product of two simplices and, more generally, of root polytopes are closely related to Gelfand-Kapranov-Zelevinsky's theory of discriminants, to tropical geometry, tropical oriented matroids, and to generalized permutohedra. We introduce a new approach to these objects, identifying a triangulation of a root polytope with a certain bijection between lattice points of two generalized permutohedra. In order to study such bijections, we define trianguloids as edge-colored graphs satisfying simple local axioms. We prove that trianguloids are in bijection with triangulations of r…

On the generalized lower bound conjecture for polytopes and spheres

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If $P$ is a simplicial $d$-polytope then its $h$-vector $(h_0,h_1,...,h_d)$ satisfies $h_0 \leq h_1 \leq ... \leq h_{\lfloor \frac d 2 \rfloor}$. Moreover, if $h_{r-1}=h_r$ for some $r \leq \frac d 2$ then $P$ can be triangulated without introducing simplices of dimension $\leq d-r$. The first part of the conjecture was solved by Stanley in 1980 using the hard Lefschetz theorem for projective toric varieties. In this paper, we give a proof of the remaining part of the conjecture…

Generalized Permutohedra from Probabilistic Graphical Models

A graphical model encodes conditional independence relations via the Markov properties. For an undirected graph these conditional independence relations can be represented by a simple polytope known as the graph associahedron, which can be constructed as a Minkowski sum of standard simplices. There is an analogous polytope for conditional independence relations coming from a regular Gaussian model, and it can be defined using multiinformation or relative entropy. For directed acyclic graphical models and also for mixed graphical models containing undirected, directed and bidirected edges, we g…