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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 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 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)

Let S be a finite set of n points in the plane in general position. We prove that every inclusion-maximal family of subsets of S separable by convex pseudo-circles has the same cardinal (n 0)+(n 1)+(n 2)+(n 3). This number does not depend on the configuration of S and is the same as the number of subsets of S separable by true circles. Buzaglo, Holzman, and… (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)