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Given a set of points called sites, the Voronoi diagram is a partition of the plane into sets of points having the same closest site. Several generalizations of the Voronoi diagram have been studied, mainly Voronoi diagrams for different distances (other than the Euclidean one), and Voronoi diagrams for sites that are not necessarily points (line segments… (More)

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 triangu-lations 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… (More)

Given a set S of n points (called sites) in a d-dimensional Euclidean space E and an integer k, 1 ≤ k ≤ n − 1, we consider three known structures that are defined through subsets of k elements of S: The k-set polytope of S, the order-k Voronoi diagram of S, and its dual, the order-k Delaunay diagram of S. We give a new compact characterization of… (More)

Given a set V of n points in the plane, no three of them being collinear, a convex inclusion chain of V is an ordering of the points of V such that no point belongs to the convex hull of the points preceding it in the ordering. We call k-set of the convex inclusion chain, every k-set of an initial subsequence of at least k points of the ordering. We show… (More)

Given a set S of disjoint line segments in the plane, which we call sites, a segment triangulation of S is a partition of the convex hull of S into sites, edges, and faces. The set of faces is a maximal set of disjoint triangles such that the vertices of each triangle are on three distinct sites. The segment Delaunay triangulation of S is the segment… (More)