Voronoi Polytope Research Articles

Logarithmic Voronoi polytopes for discrete linear models

Logarithmic Voronoi polytopes for discrete linear models

Lattice of integer flows and the poset of strongly connected orientations for regular matroids

A 2010 result of Amini provides a way to extract information about the structure of the graph from the geometry of the Voronoi polytope of the lattice of integer flows (which determines the graph up to two-isomorphism). Specifically, Amini shows that the face poset of the Voronoi polytope is isomorphic to the poset of strongly connected orientations of subgraphs. This answers a question raised by Caporaso and Viviani, and Amini also proves a dual result for integer cuts. In this paper we generalize Amini's result to regular matroids; in this context the theorem for integer cuts becomes a direct consequence of the theorem for integer flows, by making duality explicit as matroid duality.

On Combinatorics of Voronoi Polytopes for Perturbations of the Dual Root Lattices

The Voronoi conjecture on parallelohedra claims that for every convex polytope P that tiles Euclidean d-dimensional space with translations there exists a d-dimensional lattice such that P and the Voronoi polytope of this lattice are affinely equivalent. The Voronoi conjecture is still open for the general case but it is known that some combinatorial restrictions for the face structure of P ensure that the Voronoi conjecture holds for P. In this article, we prove that if P is the Voronoi polytope of one of the dual root lattices or or their small perturbations, then every parallelohedron combinatorially equivalent to P in strong sense satisfies the Voronoi conjecture.

Logarithmic Voronoi cells

We study Voronoi cells in the statistical setting by considering preimages of the maximum likelihood estimator that tessellate an open probability simplex. In general, logarithmic Voronoi cells are convex sets. However, for certain algebraic models, namely finite models, models with ML degree 1, linear models, and log-linear (or toric) models, we show that logarithmic Voronoi cells are polytopes. As a corollary, the algebraic moment map has polytopes for both its fibres and its image, when restricted to the simplex. We also compute non-polytopal logarithmic Voronoi cells using numerical algebraic geometry. Finally, we determine logarithmic Voronoi polytopes for the finite model consisting of all empirical distributions of a fixed sample size. These polytopes are dual to the logarithmic root polytopes of Lie type A, and we characterize their faces.

A tool for pathline creation using TOUGH simulation results and fully unstructured 3D Voronoi grids

Pathline or streamline computation is used today to improve the understanding of fluid flow behavior in groundwater modelling. In particular, pathline computation on unstructured grids requires an accurate velocity interpolation method when the flow velocity field is given by numerical simulators only at the faces of the grid blocks; as for example in the case of the Integral Finite Difference method is used for spatial discretization. In the case of 3D Voronoi unstructured grids, a corner velocity interpolation (CVI) method allows continuous velocity field interpolation while preserving the mass conservation principle. We have developed a computer code, named TOUGH2Path, able to compute the velocity field based on the CVI method and generalized barycentric coordinates for 3D Voronoi polytope grid blocks tailored for the TOUGH family of codes. Results for the classical quarter five spot problem show that the coded algorithm correctly reproduces the pathlines computed with semi-analytical methods.

Voronoi polytopes for polyhedral norms on lattices

A polyhedral norm is a norm N on Rn for which the set N(x)≤1 is a polytope. This covers the case of the L1 and L∞ norms. We consider here effective algorithms for determining the Voronoi polytope for such norms with a point set being a lattice. The algorithms, that we propose, use the symmetries effectively in order to compute a decomposition of the space into convex polytopes named VN-spaces. The Voronoi polytopes and other geometrical information are easily obtained from it.

Covering a convex 3D polytope by a minimal number of congruent spheres

The problem of covering a convex 3D polytope by the minimal number of congruent spheres is reduced to a sequence of problems of minimising sphere radius when fixing the number of the spheres. We form a mathematical model of the problem using the Voronoi polytopes. Characteristics of the model are investigated. Extrema are attained at the vertices of the Voronoi polytopes constructed for sphere centres. To search for local minima, a modification of the Zoutendijk feasible directions method in a combination with random search is developed. Some numerical results for a cube and a non-regular octahedron are obtained.

The geometry and combinatorics of cographic toric face rings

In this paper we define and study a ring associated to a graph that we call the cographic toric face ring, or simply the cographic ring. The cographic ring is the toric face ring defined by the following equivalent combinatorial structures of a graph: the cographic arrangement of hyperplanes, the Voronoi polytope, and the poset of totally cyclic orientations. We describe the properties of the cographic ring and, in particular, relate the invariants of the ring to the invariants of the corresponding graph. Our study of the cographic ring fits into a body of work on describing rings constructed from graphs. Among the rings that can be constructed from a graph, cographic rings are particularly interesting because they appear in the study of compactified Jacobians of nodal curves.

The Minkowski sum of a zonotope and the Voronoi polytope of the root lattice

We show that the Minkowski sum of the Voronoi polytope of the root lattice and the zonotope is a 7-dimensional parallelohedron if and only if the set consists of minimal vectors of the dual lattice up to scalar multiplication, and does not contain forbidden sets. The minimal vectors of are the vectors of the classical root system . If the -norm of the roots is set equal to 2, then the scalar products of minimal vectors from the dual lattice only take the values . A set of minimal vectors is referred to as forbidden if it consists of six vectors, and the directions of some of these vectors can be changed so as to obtain a set of six vectors with all the pairwise scalar products equal to .Bibliography: 11 titles.

Delaunay and voronoi polytopes of the root lattice E 7 and of the dual lattice E*7

We give a detailed geometrically clear description of all faces of the Delaunay and Voronoi polytopes of the root lattice E7 and the dual lattice E*7. Here three uniform polytopes related to the Coxeter-Dynkin diagram of the Lie algebra E7 play a special role. These are the Gosset polytope PGos = 321, which is a Delaunay polytope, the contact polytope 231 (both for the lattice E7), and the Voronoi polytope PV(E*7) = 132 of the dual lattice E*7. This paper can be considered as an illustration of the methods for studying Delaunay and Voronoi polytopes.

Properties of parallelotopes equivalent to Voronoi’s conjecture

A parallelotope is a polytope whose translation copies fill space without gaps and intersections by interior points. Voronoi conjectured that each parallelotope is an affine image of the Dirichlet domain of a lattice, that is to say a Voronoi polytope. We give several properties of a parallelotope and prove that each of them is equivalent to it being an affine image of a Voronoi polytope.

Covering Presentation and Coloring of Dual Canonical Tilings

In the crystallography of quasicrystals, the space for the structure description is taken as a subspace E‖, embedded irrationally in an N-dimensional space equipped with a lattice Λ. Quasiperiodic dual canonical tilings (\(\mathcal{T}\), Λ), (\(\mathcal{T}\)*, Λ) arise from the geometry and projection of Voronoi and Delone polytopes, respectively, associated with Λ. In a new approach to quasicrystals, tiles are replaced by a few covering clusters. As a typical example, the tiling (\(\mathcal{T}\)*, A4) of fivefold symmetry is covered by two Delone clusters. This construction is called a covering presentation compatible with the tiling. We show that any given tile in the tiling can be uniquely associated with a specific overlap of Delone clusters. Each specific overlap is called a color for this tile. The range of colors and their relative frequencies are explicitly determined.

Non-rigidity Degree of a Lattice and Rigid Lattices

Voronoi defines a partition of the cone of positive semidefinite n -ary formsPn into L -type domains. Each L -type domain is an open polyhedral cone of dimensionk , 1 ≤k≤n(n+ 1)2, where n is the number of variables and dimension of the corresponding lattice. We define a non-rigidity degree of a lattice as the dimension of the L -type domain containing the lattice. We prove that the non-rigidity degree of a lattice equals the corank of a system of equalities connecting norms of minimal vectors of cosets of 2 L in L. A lattice of non-rigidity degree 1 is called rigid. A lattice is rigid if any of its sufficiently small deformations distinct from a homothety changes its L -type. Using the list of 84 zone-contracted Voronoi polytopes inR5given by Engel [8], we give a complete list of seven five-dimensional rigid lattices.

Zonotopes, Dicings, and Voronoi’s Conjecture on Parallelohedra

In 1909, Voronoi conjectured that if some selection of translates of a polytope forms a facet-to-facet tiling of euclidean space, then the polytope is affinely equivalent to the Voronoi polytope for a lattice. He referred to polytopes with this tiling property as parallelohedra, but they are now frequently called parallelotopes. I show that Voronoi’s conjecture holds for the special case where the parallelotope is a zonotope. I also show that the Voronoi polytope for a lattice is a zonotope if and only if the Delaunay tiling for the lattice is a dicing (defined at the beginning of Section 3).

Random exploration of the three regular polytopes

There are just three regular polytopes (the hypercube, cross polytope and the simplex) in Euclidean (n>4) space. We calculate their dimensions, including the distance from the centroid to the periphery in a random direction-that of a white Gaussian-noise vector. As n/spl rarr//spl infin/, this distance becomes very predictable. It differs from the distance near which almost all of the volume and surface of the polytope lie.

Radial generation of n-dimensional poisson processes

A simple method is proposed for the generation of successive ‘nearest neighbours' to a given origin in ann-dimensional Poisson process. It is shown that the method provides efficient simulation of random Voronoi polytopes. Results are given of simulation studies in two and three dimensions.

Radial generation of n-dimensional poisson processes

A simple method is proposed for the generation of successive ‘nearest neighbours' to a given origin in an n-dimensional Poisson process. It is shown that the method provides efficient simulation of random Voronoi polytopes. Results are given of simulation studies in two and three dimensions.

Nearest neighbors and Voronoi regions in certain point processes

We investigate, for several models of point processes, the (random) number N of points which have a given point as their nearest neighbor. The largedimensional limit of Poisson processes is treated by considering for n points independently and uniformly distributed in a d-dimensional cube of volume n and showing that Poisson (λ= 1). An asymptotic Poisson (λ= 1) distribution also holds for many of the other models. On the other hand, we find that . Related results concern the (random) volume, , of a Voronoi polytope (or Dirichlet cell) in the cube model; we find that while

Nearest neighbors and Voronoi regions in certain point processes

We investigate, for several models of point processes, the (random) number N of points which have a given point as their nearest neighbor. The largedimensional limit of Poisson processes is treated by considering for n points independently and uniformly distributed in a d-dimensional cube of volume n and showing that Poisson (λ= 1). An asymptotic Poisson (λ= 1) distribution also holds for many of the other models. On the other hand, we find that . Related results concern the (random) volume, , of a Voronoi polytope (or Dirichlet cell) in the cube model; we find that while