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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 Pinchasi showed that it is an upper bound for the number of subsets separable by (non necessarily convex) pseudo-circles. Actually, we first count the number of elements in a maximal family of k-subsets of S separable by convex pseudo-circles, for a given k. We show that Lee's result on the number of k-subsets separable by true circles still holds for convex pseudo-circles. In particular, this means that the number of k-subsets of S separable by a maximal family of convex pseudo-circles is an invariant of S: It does not depend on the choice of the maximal family. To prove this result, we introduce a graph that generalizes the dual graph of the order-k Voronoi diagram, and whose vertices are the k-subsets of S separable by a maximal family of convex pseudo-circles. In order to count the number of vertices of this graph, we first show that it admits a planar realization which is a triangulation. It turns out (but is not detailed in the present paper) that these triangulations are the centroid triangulations Liu and Snoeyink conjectured to construct. Theorem 1.2. Let S be a set of n points in the plane in general position and k ∈ {1,. .. , n}. Every inclusion-maximal family of k-subsets of S separable by convex pseudo-circles admits 2kn − n − k 2 + 1 − k−1 i=1 a i (S) elements. To prove these results, we first characterize convex pseudo-circle separability in terms of convex hulls (Section 2). We show that a family F of subsets of S is separable by convex pseudo-circles if conv(T) ∩ S = T for all T ∈ F, and conv(T \ T ′) ∩ conv(T ′ \ T) = ∅ for all T, T ′ ∈ F. We also characterize in a similar way families of pairs (P, Q) of subsets of S determined by convex pseudo-circles passing through the points of Q, containing P , and excluding S \ (P ∪ Q). Such pairs are called convex pairs. We then introduce a graph that generalizes the dual of the order-k Voronoi diagram (Sections 3 and 4). The vertices of this graph are the elements of size k in a family of subsets of S separable by a maximal family C of convex pseudo-circles. The edges of the graph are the convex pairs of the form (P, {s, t}), with |P | = k − 1, determined by convex pseudo-circles compatible with C. The edge (P, {s, t}) connects the two vertices P ∪ {s} and P ∪ {t}. The key result of the article, Theorem 3.9, is that this graph admits a planar geometric realization which induces a triangulation. Proposition 3.4 already shows that the edges of this realization are disjoint. The main difficulty is to show that every edge is incident to a triangle, see Theorem 3.7. The long and tricky proof of this result can be found in Section 5. Two important contributions of our work are, firstly, the notion of convex p...

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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 for example).In this paper we present a new generalization of the Voronoi diagram in the plane, in which we shift our interest from points to lines, that is, we compute the partition of the set of lines in the plane into sets of lines having the same closest site (where sites are points in the plane). We first define formally this diagram and give first properties. Then we use a duality relationship between points and lines to visualize this data structure and give more properties. We show that the size of this line space Voronoi diagram for n sites is in Θ(n 2 ) and give an optimal algorithm for its explicit computation.We then show a remarkable relationship between this diagram and the dual arrangement of the sites and give a new result on an arrangement of lines: we show that the size of the zone of a line augmented with its incident faces is still in O(n). We finally apply this result to show that the size of the zone of a line in the line space Voronoi diagram is in O(n).

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