Geometric triangulations and discrete Laplacians on manifolds

The Voronoi Functional is Maximized by the Delaunay Triangulation in the Plane

We introduce the Voronoi functional of a triangulation of a finite set of points in the Euclidean plane and prove that among all geometric triangulations of the point set, the Delaunay triangulation maximizes the functional. This result neither extends to topological triangulations in the plane nor to geometric triangulations in three and higher dimensions.

Convex polygons in geometric triangulations

We show that the maximum number of convex polygons in a triangulation of $n$ points in the plane is $O(1.5029^n)$. This improves an earlier bound of $O(1.6181^n)$ established by van Kreveld, Löffler, and Pach (2012) and almost matches the current best lower bound of $Ω(1.5028^n)$ due to the same authors. Given a planar straight-line graph $G$ with $n$ vertices, we show how to compute efficiently the number of convex polygons in $G$.

Old and new geometric polyhedra with few vertices

This paper deals with triangulations of the 2-torus with the vertex labeled general octahedral graph $O_4$ which is isomorphic to the complete four-partite graph $K_{2,2,2,2}$; it is known that there exist precisely twelve such triangulations. We find all the 12 triangulations in a Schlegel diagram of the hyperoctahedron and realize all of them geometrically with the same 1-skeleton in 3-space. In particular, we identify two geometric polyhedral tori (both without self-intersections) with the same 1-skeleton in 3-space, but without a single common face, or in other words their intersection (as…

A general discrete Wirtinger inequality and spectra of discrete Laplacians

We prove an inequality that generalizes the Fan-Taussky-Todd discrete analog of the Wirtinger inequality. It is equivalent to an estimate on the spectral gap of a weighted discrete Laplacian on the circle. The proof uses a geometric construction related to the discrete isoperimetric problem on the surface of a cone. In higher dimensions, the mixed volumes theory leads to similar results, which allows us to associate a discrete Laplace operator to every geodesic triangulation of the sphere and, by analogy, to every triangulated spherical cone-metric. For a cone-metric with positive singular cur…

Non-Realizable Minimal Vertex Triangulations of Surfaces: Showing Non-Realizability using Oriented Matroids and Satisfiability Solvers

We show that no minimal vertex triangulation of a closed, connected, orientable 2-manifold of genus 6 admits a polyhedral embedding in R^3. We also provide examples of minimal vertex triangulations of closed, connected, orientable 2-manifolds of genus 5 that do not admit any polyhedral embeddings. We construct a new infinite family of non-realizable triangulations of surfaces. These results were achieved by transforming the problem of finding suitable oriented matroids into a satisfiability problem. This method can be applied to other geometric realizability problems, e.g. for face lattices of…

离散 Laplace 算子——球面与双曲情形

Discrete Laplacians -- spherical and hyperbolic

本文定义球面与双曲三角曲面上的离散 Laplace 算子，保持 Delaunay 非负权、面积表示、共形特征函数及无穷小等距变形等几何结构。