相关论文: Voronoi Polytopes for Polyhedral Norms on Lattices
For a measure on a subset of the complex plane we consider $L^p$-optimal weighted polynomials, namely, monic polynomials of degree $n$ with a varying weight of the form $w^n = {\rm e}^{-n V}$ which minimize the $L^p$-norms, $1 \leq p \leq…
A set of orthogonal polynomials on the unit disk $B(0,1)$ known as Zernike polynomials are commonly used in the analysis and evaluation of optical systems. Here Zernike polynomials are used to construct wavelets for polynomial subspaces of…
We consider the large-$N$ asymptotics of a system of discrete orthogonal polynomials on an infinite regular lattice of mesh $\frac{1}{N}$, with weight $e^{-NV(x)}$, where $V(x)$ is a real analytic function with sufficient growth at…
In (the surface of) a convex polytope P^n in R^n+1, for small prescribed volume, geodesic balls about some vertex minimize perimeter. This revision corrects a mistake in the mass bound argument in the proof of Theorem 3.8.
This paper introduces an algebraic combinatorial approach to simplicial cone decompositions, a key step in solving inhomogeneous linear Diophantine systems and counting lattice points in polytopes. We use constant term manipulation on the…
Polyhedral meshes are increasingly becoming an attractive option with particular advantages over traditional meshes for certain applications. What has been missing is a robust polyhedral meshing algorithm that can handle broad classes of…
Effective transport properties of heterogeneous structures are predicted by geometric microstructural parameters, but these can be difficult to calculate. Here, a boundary element code with a recurrent series method accurately and…
A convex polygon is defined as a sequence (V_0,...,V_{n-1}) of points on a plane such that the union of the edges [V_0,V_1],..., [V_{n-2},V_{n-1}], [V_{n-1},V_0] coincides with the boundary of the convex hull of the set of vertices…
We define Q-normal lattice polytopes. Natural examples of such polytopes are Cayley sums of strictly combinatorially equivalent lattice polytopes, which correspond to particularly nice toric fibrations, namely toric projective bundles. In a…
We investigate symmetric edge polytopes generated by Erd\H{o}s--R\'enyi random graphs in a high-dimensional regime. These objects provide a natural and largely unexplored model of random lattice polytopes, in which geometric properties are…
The Voronoi diagram is a certain geometric data structure which has numerous applications in various scientific and technological fields. The theory of algorithms for computing 2D Euclidean Voronoi diagrams of point sites is rich and…
We study the perfect matching lattice of a matching covered graph $G$, generated by the incidence vectors of its perfect matchings. Building on results of Lov\'asz and de Carvalho, Lucchesi, and Murty, we give a polynomial-time algorithm…
A lattice Delaunay polytope D is called perfect if it has the property that there is a unique circumscribing ellipsoid with interior free of lattice points, and with the surface containing only those lattice points that are the vertices of…
We give an incremental polynomial time algorithm for enumerating the vertices of any polyhedron $\mathcal{P}(A,\mathbf{1})=\{x\in\RR^n \mid Ax\geq \b1,~x\geq \b0\}$, when $A$ is a totally unimodular matrix. Our algorithm is based on…
Equivariant Ehrhart theory generalizes the study of lattice point enumeration to also account for the symmetries of a polytope under a linear group action. We present a catalogue of techniques with applications in this field, including…
Given a set of point sites in a simple polygon, the geodesic farthest-point Voronoi diagram partitions the polygon into cells, at most one cell per site, such that every point in a cell has the same farthest site with respect to the…
In nonequilibrium statistical physics, quantifying the nearest (and higher-order) neighbors and free volumes of particles in many-body systems is crucial to elucidating the origin of macroscopic collective phenomena, such as glass/granular…
Estimating the number of vertices of a two dimensional projection, called a shadow, of a polytope is a fundamental tool for understanding the performance of the shadow simplex method for linear programming among other applications. We prove…
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…
In this thesis we study sets of points in the plane and their Voronoi diagrams, in particular when the points coincide. We bring together two ways of studying point sets that have received a lot of attention in recent years: Voronoi…