Related papers: Properties of Polytopes Representing Natural Numbe…
The polymake software system deals with convex polytopes and related objects from geometric combinatorics. This note reports on a new implementation of a subclass for lattice polytopes. The features displayed are enabled by recent changes…
Polytope numbers for a polytope are a sequence of nonnegative integers that are defined by the facial information of a polytope. Every polygon is triangulable and a higher dimensional analogue of this fact states that every polytope is…
We describe the computation of polytope volumes by descent in the face lattice, its implementation in Normaliz, and the connection to reverse-lexicographic triangulations. The efficiency of the algorithm is demonstrated by several high…
Preorder polytopes, defined from preorders on finite sets, are introduced and studied from a lattice point enumeration point of view. They naturally generalize arbor polytopes, recently introduced and studied by the second named author.…
We provide two algorithms for computing the volume of a convex polytope with half-space representation {x>=0; Ax <=b} for some (m,n) matrix A and some m-vector b. Both algorithms have a O(n^m) computational complexity which makes them…
The main purpose of this paper is to report on the state of the art of computing integer hulls and their facets as well as counting lattice points in convex polytopes. Using the polymake system we explore various algorithms and…
We study the harmonic polytope, which arose in Ardila, Denham, and Huh's work on the Lagrangian geometry of matroids. We describe its combinatorial structure, showing that it is a $(2n-2)$-dimensional polytope with…
We develop a collection of numerical algorithms which connect ideas from polyhedral geometry and algebraic geometry. The first algorithm we develop functions as a numerical oracle for the Newton polytope of a hypersurface and is based on…
The mathematical software system polymake provides a wide range of functions for convex polytopes, simplicial complexes, and other objects. A large part of this paper is dedicated to a tutorial which exemplifies the usage. Later sections…
In the present article, real number representations, that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are…
We present a new algorithm for computing the volume of an arbitrary matroid base polytope. We provide two applications of this approach: a relation between the volume of the base polytope of a matroid $\M$ and its relaxation $\M'$, and a…
We prove that each bounded polytope can be represented as a polynomial zonotope, which we refer to as the Z-representation of polytopes. Previous representations are the vertex representation (V-representation) and the halfspace…
Computing mixed volume of convex polytopes is an important problem in computational algebraic geometry. This paper establishes sufficient conditions under which the mixed volume of several convex polytopes exactly equals the normalized…
This is a survey on algorithmic questions about combinatorial and geometric properties of convex polytopes. We give a list of 35 problems; for each the current state of knowledege on its theoretical complexity status is reported. The…
In this paper we discuss about properties of lattices and its application in theoretical and algorithmic number theory. This result of Minkowski regarding the lattices initiated the subject of Geometry of Numbers, which uses geometry to…
We introduce a novel intrinsic volume concept in tropical geometry. This is achieved by developing the foundations of a tropical analog of lattice point counting in polytopes. We exhibit the basic properties and compare it to existing…
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…
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 present an extremely elementary construction of the simple Lie algebras over the complex numbers in all of their minuscule representations, using the vertices of various polytopes. The construction itself requires no complicated…
We survey the computation of polytope volumes by the algorithms of Normaliz to which the Lawrence algorithm has recently been added. It has enabled us to master volume computations for polytopes from social choice in dimension $119$. This…