Related papers: Voronoi Polytopes for Polyhedral Norms on Lattices

A comprehensive overview of lattice rules and polynomial lattice rules is given for function spaces based on $\ell_p$ semi-norms. Good lattice rules and polynomial lattice rules are defined as those obtaining worst-case errors bounded by…

We investigate the combinatorial complexity of geodesic Voronoi diagrams on polyhedral terrains using a probabilistic analysis. Aronov etal [ABT08] prove that, if one makes certain realistic input assumptions on the terrain, this complexity…

In this paper, we give pointwise estimates of a Vorono\"i-based finite volume approximation of the Laplace-Beltrami operator on Vorono\"i-Delaunay decompositions of the sphere. These estimates are the basis for a local error analysis, in…

In this paper for any dimension n we give a complete list of lattice convex polytopes in R^n that are regular with respect to the group of affine transformations preserving the lattice.

Giving a joint generalization of a result of Brazitikos, Chasapis and Hioni and results of Giannopoulos and Milman, we prove that roughly $\left\lceil \frac{d}{(1-\vartheta)^d}\ln\frac{1}{(1-\vartheta)^d} \right\rceil$ points chosen…

We introduce polyhedra circuits. Each polyhedra circuit characterizes a geometric region in $\mathbb{R}^d$. They can be applied to represent a rich class of geometric objects, which include all polyhedra and the union of a finite number of…

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…

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…

We present two algorithms that compute the Newton polytope of a polynomial defining a hypersurface H in C^n using numerical computation. The first algorithm assumes that we may only compute values of f - this may occur if f is given as a…

We extend the theory of logarithmic Voronoi cells to Gaussian statistical models. In general, a logarithmic Voronoi cell at a point on a Gaussian model is a convex set contained in its log-normal spectrahedron. We show that for models of ML…

For a given lattice, we establish an equivalence involving a closed zone of the corresponding Voronoi polytope, a lamina hyperplane of the corresponding Delaunay partition and a quadratic form of rank 1 being an extreme ray of the…

A polynomial knot in $\mathbb{R}^n$ is a smooth embedding of $\mathbb{R}$ in $\mathbb{R}^n$ such that the component functions are real polynomials. In the earlier paper with Mishra, we have studied the space $\mathcal{P}$ of polynomial…

We describe an algorithm for determining whether two convex polytopes P and Q, embedded in a lattice, are isomorphic with respect to a lattice automorphism. We extend this to a method for determining if P and Q are equivalent, i.e. whether…

In this paper we obtain $L^1$-weighted norms of classical orthogonal polynomials (Hermite, Laguerre and Jacobi polynomials) in terms of the zeros of these orthogonal polynomials; these expressions are usually known as quadrature rules. In…

Extraction of a high-fidelity 3D medial axis is a crucial operation in CAD. When dealing with a polygonal model as input, ensuring accuracy and tidiness becomes challenging due to discretization errors inherent in the mesh surface.…

We present two methods to continuously and piecewise-linearly parametrize rank-3 lattices by vectors of $\RR^{13}$, which provides an efficient way to judge if two sets of parameters provide nearly identical lattices within their margins of…

In a nutshell, we show that polynomials and nested polytopes are topological, algebraic and algorithmically equivalent. Given two polytops $A\subseteq B$ and a number $k$, the Nested Polytope Problem (NPP) asks, if there exists a polytope…

The intersections between a spherical shell and the faces of Voronoi's polyhedrons are numerically evaluated. The nodes of these intersections are the points that share the same distances from three nuclei. The nodes are assumed to be the…

We introduce machine learning methodology to the study of lattice polytopes. With supervised learning techniques, we predict standard properties such as volume, dual volume, reflexivity, etc, with accuracies up to 100%. We focus on 2d…

This paper is devoted to the study the $m$-point homogeneity property and the point homogeneity degree for finite metric spaces. Since the vertex sets of regular polytopes, as well as of some their generalizations, are homogeneous, we pay…