Separation by Convex Pseudo-Circles

Abstract

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...

Algebraic $k$-sets and generally neighborly embeddings

“…An essentially equivalent result was proven earlier in [4] by counting vertices of certain Voronoi diagrams. [7] extended the result to convex pseudo-circles.…”

“…It is natural to ask similar questions for families of surfaces different from all hyperplanes. These sorts of questions have been studied in [4,5,6,7]. [5], of particular interest to us, shows that for any set of 2n + 1 points in general position in the plane, the number of circles that go through 3 points and split the remaining points in half is exactly n 2 .…”

Tetrahedral and pentahedral cages for discs

“…Proposition 3.6 (Chevallier, Fruchard [5]). For any (bounded) combinatorial prism with triangular faces ∆ and ∆ , it is impossible that ∆ lies in the interior of a prism associated with ∆ , and ∆ lies in the interior of a prism associated with ∆.…”

On the Number of Hyperedges in the Hypergraph of Lines and Pseudo-Discs

“…The family of hypergraphs H(P, F) -for a general F and in the special case where all elements of F are convex -have been studied extensively (see, e.g., [1,3,6,9,13]). In particular, it was proved in [7] that for any P, F, the Delaunay graph of H(P, F) (namely, the restriction of H to hyperedges of size 2) is planar, and that for any fixed t, the number of hyperedges of H(P, F) of size t is bounded by O(t 2 |P |).…”