In this short paper we consider the functional density on sets of uniformly bounded triangulations with fixed sets of vertices. We prove that if a functional attains its minimum on the Delaunay triangulation for every finite set in the plane, then for infinite sets the density of this functional attains its minimum also on the Delaunay triangulations. A Delaunay set in E is a set of points X fo...

Given a set S of line segments in the plane, we introduce a new family of partitions of the convex hull of S called segment triangulations of S. The set of faces of such a triangulation is a maximal set of disjoint triangles that cut S at, and only at, their vertices. Surprisingly, several properties of point set triangulations extend to segment triangulations. Thus, the number of their faces i...

In the study of depth functions it is important to decide whether we want such a function to be sensitive to multimodality or not. In this paper we analyze the Delaunay depth function, which is sensitive to multimodality and compare this depth with others, as convex depth and location depth. We study the stratification that Delaunay depth induces in the point set (layers) and in the whole plane...

This article studies the construction of self-stabilizing topologies for distributed systems. While recent research has focused on chain topologies where nodes need to be linearized with respect to their identifiers, we explore a natural and relevant 2-dimensional generalization. In particular, we present a local self-stabilizing algorithm DStab which is based on the concept of ‘‘local Delaunay...

We consider an extension of the triangular-distance Delaunay graphs (TD-Delaunay) on a set P of points in general position in the plane. In TD-Delaunay, the convex distance is defined by a fixed-oriented equilateral triangle 5, and there is an edge between two points in P if and only if there is an empty homothet of 5 having the two points on its boundary. We consider higher-order triangular-di...

The paper presents a new robust method for finding points of intersection of line segments in the plane. In this method the subdivision of the plane based on the Delaunay triangulation plays the main role. First, the Delaunay triangulation spanning the end points of line segments is constructed. Next, for line segments that are not realized by Delaunay edges, midpoints are inserted recursively ...

Four constructions of constant mean curvature (CMC) hypersurfaces in Sn+1 are given, which should be considered analogues of ‘classical’ constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an equator, are shown to exist for all the values of the mean curvature. S...