Related papers: Embedding Stacked Polytopes on a Polynomial-Size G…
Many results in mass partitions are proved by lifting $\mathbb{R}^d$ to a higher-dimensional space and dividing the higher-dimensional space into pieces. We extend such methods to use lifting arguments to polyhedral surfaces. Among other…
Given a (finite) simplicial complex, we define its $i$-th Laplacian polytope as the convex hull of the columns of its $i$-th Laplacian matrix. This extends Laplacian simplices of finite simple graphs, as introduced by Braun and Meyer. After…
A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least $2k$ vertices is \textit{$k$-linked} if,…
Neural networks are widely used as a model for classification in a large variety of tasks. Typically, a learnable transformation (i.e. the classifier) is placed at the end of such models returning a value for each class used for…
A multidimensional nonnegative matrix is called polystochastic if the sum of entries in each of its lines equals $1$. The set of all polystochastic matrices of order $n$ and dimension $d$ is a convex polytope $\Omega_n^d$ known as the…
We investigate a novel setting for polytope rigidity, where a flex must preserve edge lengths and the planarity of faces, but is allowed to change the shapes of faces. For instance, the regular cube is flexible in this notion. We present…
A prismatoid is a polytope with all its vertices contained in two parallel facets, called its bases. Its width is the number of steps needed to go from one base to the other in the dual graph. The author recently showed in arXiv:1006.2814…
We demonstrate a novel route of spatially organizing the colloid arrangements on the polymer-grafted substrate by use of self-consistent field and density functional theories. We find that grafting of polymers onto a substrate can…
A prismatoid is a polytope with all its vertices contained in two parallel facets, called its bases. Its width is the number of steps needed to go from one base to the other in the dual graph. The first author recently showed that the…
We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-join and skew…
Symmetric edge polytopes are a recent and well-studied family of centrally symmetric polytopes arising from graphs. In this paper, we introduce a generalization of this family to arbitrary simplicial complexes. We show how topological…
Given a finite collection P of convex n-polytopes in RP^n (n>1), we consider a real projective manifold M which is obtained by gluing together the polytopes in P along their facets in such a way that the union of any two adjacent polytopes…
A construction of polytopes is given based on integers. These geometries are constructed through a mapping to pure numbers and have multiple applications, including statistical mechanics and computer science. The number form is useful in…
We characterize the polytopes in $\mathbb{R}^d$ (not necessarily convex or connected ones) which multi-tile the space by translations along a given lattice. We also give a necessary and sufficient condition for two polytopes in…
If we fix the angles at the vertices of a convex planar $n$-gon, the lengths of its edges must satisfy two linear constraints in order for it to close up. If we also require unit perimeter, our vectors of $n$ edge lengths form a convex…
Given a set $S \subseteq \mathbb{R}^d$, a hollow polytope has vertices in $S$ but contains no other point of $S$ in its interior. We prove upper and lower bounds on the maximum number of vertices of hollow polytopes whose facets are…
In the Minimum $d$-Dimensional Arrangement Problem (d-dimAP) we are given a graph with edge weights, and the goal is to find a 1-1 map of the vertices into $\mathbb{Z}^d$ (for some fixed dimension $d\geq 1$) minimizing the total weighted…
The isotropy constant of any $d$-dimensional polytope with $n$ vertices is bounded by $C \sqrt{n/d}$ where $C>0$ is a numerical constant.
The degree partition of a simple graph is its degree sequence rearranged in weakly decreasing order. The polytope of degree partitions (respectively, degree sequences) is the convex hull of all degree partitions (respectively, degree…
We present a novel method to perform numerical integration over curved polyhedra enclosed by high-order parametric surfaces. Such a polyhedron is first decomposed into a set of triangular and/or rectangular pyramids, whose certain faces…