Neighborliness of Marginal Polytopes

On the neighborliness of dual flow polytopes of quivers

In this note we investigate under which conditions the dual of the flow polytope (henceforth referred to as the `dual flow polytope') of a quiver is k-neighborly, for generic weights near the canonical weight. We provide a lower bound on k, depending only on the edge connectivity of the underlying graph, for such weights. In the case where the canonical weight is in fact generic, we explicitly determine a vertex presentation of the dual flow polytope. Finally, we specialize our results to the case of complete, bipartite quivers where we show that the canonical weight is indeed generic, and are…

Local neighborliness of the symmetric moment curve

A centrally symmetric analogue of the cyclic polytope, the bicyclic polytope, was defined in [BN08]. The bicyclic polytope is defined by the convex hull of finitely many points on the symmetric moment curve where the set of points has a symmetry about the origin. In this paper, we study the Barvinok-Novik orbitope, the convex hull of the symmetric moment curve. It was proven in [BN08] that the orbitope is locally $k$-neighborly, that is, the convex hull of any set of $k$ distinct points on an arc of length not exceeding $ϕ_k$ in $\mathbb{S}^1$ is a $(k-1)$-dimensional face of the orbitope for …

A tale of centrally symmetric polytopes and spheres

This paper is a survey of recent advances as well as open problems in the study of face numbers of centrally symmetric simplicial polytopes and spheres. The topics discussed range from neighborliness of centrally symmetric polytopes and the upper bound theorem for centrally symmetric simplicial spheres to the generalized lower bound theorem for centrally symmetric simplicial polytopes and the lower bound conjecture for centrally symmetric simplicial spheres and manifolds.

二元层次模型的广义割多面体

Generalized Cut Polytopes for Binary Hierarchical Models

本文引入广义割多面体，通过广义协方差映射与任意单纯复形的二元边缘多面体同构，并给出若干重要单纯复形族的完整H-表示及部分体积计算。

层次模型的码度、弱最大似然阈值与格森斯坦性质

The Codegree, Weak Maximum Likelihood Threshold, and the Gorenstein Property of Hierarchical Models

本文研究了层次模型的码度、弱最大似然阈值及格森斯坦性质，提出了码度的下界和猜想公式，并探讨了边际多面体的格森斯坦分类。

An Implicitization Challenge for Binary Factor Analysis

We use tropical geometry to compute the multidegree and Newton polytope of the hypersurface of a statistical model with two hidden and four observed binary random variables, solving an open question stated by Drton, Sturmfels and Sullivant in "Lectures on Algebraic Statistics" (Problem 7.7). The model is obtained from the undirected graphical model of the complete bipartite graph $K_{2,4}$ by marginalizing two of the six binary random variables. We present algorithms for computing the Newton polytope of its defining equation by parallel walks along the polytope and its normal fan. In this way …