Related papers: On interior polytope number sequences
The aim of this paper is to study alcoved polytopes, which are polytopes arising from affine Coxeter arrangements. This class of convex polytopes includes many classical polytopes, for example, the hypersimplices. We compare two…
The interior polynomial was originally defined for hypergraphs and later shown to coincide with the Ehrhart polynomial of the root polytope of an associated bipartite graph. In previous work, we derived an alternating cycle recursion…
Let $2\le k\le d-1$ and let $P$ and $Q$ be two convex polytopes in ${\mathbb E^d}$. Assume that their projections, $P|H$, $Q|H$, onto every $k$-dimensional subspace $H$, are congruent. In this paper we show that $P$ and $Q$ or $P$ and $-Q$…
Consider a real point configuration $\mathbf{A}$ of size $n$ and an integer $r \leq n$. The vertices of the $r$-lineup polytope of $\mathbf{A}$ correspond to the possible orderings of the top $r$ points of the configuration obtained by…
In this paper, we address the problem of counting integer points in a rational polytope described by $P(y) = \{ x \in \mathbb{R}^m \colon Ax = y, x \geq 0\}$, where $A$ is an $n \times m$ integer matrix and $y$ is an $n$-dimensional integer…
Bisztriczky defines a multiplex as a generalization of a simplex, and an ordinary polytope as a generalization of a cyclic polytope. This paper presents results concerning the combinatorics of multiplexes and ordinary polytopes. The flag…
A permutation polytope is the convex hull of a group of permutation matrices. In this paper we investigate the combinatorics of permutation polytopes and their faces. As applications we completely classify permutation polytopes in…
This is an overview of results from our experiment of merging two seemingly unrelated disciplines - higher algebraic K-theory of rings and the theory of lattice polytopes. The usual K-theory is the ``theory of a unit simplex''. A conjecture…
A biconvex polytope is a classical and tropical convex hull of finitely many points. Given a biconvex polytope, for each vertex of it we construct a directed bigraph and a gammoid so that the collection of base polytopes of those gammoids…
To every integer monic polynomial of degree m can be associated m integer sequences having interesting properties to the roots of the polynomial. These sequences can be used to find the real roots of any integer monic polynomial by using…
The close relationship among the polynomial functions and Fibonacci numerical sequences is shown in this paper. These numerical sequences are defined by the recurrence equation $x_{k + n} = \displaystyle\sum_{j = 0}^{n-1}\alpha_j x_{k +…
The polytope containment problem is deciding whether a polytope is a contained within another polytope. This problem is rooted in computational convexity, and arises in applications such as verification and control of dynamical systems. The…
Quickly convergent series are given to compute polyzeta numbers. The formula involves an intricate combination of (generalized) polylogarithms at 1/2. However, the combinatorics has a very simple geometric interpretation: it corresponds…
These lectures on the combinatorics and geometry of 0/1-polytopes are meant as an \emph{introduction} and \emph{invitation}. Rather than heading for an extensive survey on 0/1-polytopes I present some interesting aspects of these objects;…
Network coding is studied when an adversary controls a subset of nodes in the network of limited quantity but unknown location. This problem is shown to be more difficult than when the adversary controls a given number of edges in the…
Let $f(x_1,...,x_k)$ be a polynomial over a field $K$. This paper considers such questions as the enumeration of the number of nonzero coefficients of $f$ or of the number of coefficients equal to $\alpha\in K^*$. For instance, if $K=\ff_q$…
Lattice polytopes are called IDP polytopes if they have the integer decomposition property, i.e., any lattice point in a $k$th dilation is a sum of $k$ lattice points in the polytope. It is a long-standing conjecture whether the numerator…
Polytope complexes are the generalisation of polygon meshes in geo-information systems (GIS) to arbitrary dimension, and a natural concept for accessing spatio-temporal information. Complexes of each dimension have a straight-forward…
For a polynomial P, we consider the sequence of iterated integrals of ln P(x). This sequence is expressed in terms of the zeros of P(x). In the special case of ln(1 + x^2), arithmetic properties of certain coefficients arising are…
A transportation polytope consists of all multidimensional arrays or tables of non-negative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have…