Density of Range Capturing Hypergraphs

For a finite set $X$ of points in the plane, a set $S$ in the plane, and a positive integer $k$, we say that a $k$-element subset $Y$ of $X$ is captured by $S$ if there is a homothetic copy $S'$ of $S$ such that $X\cap S' = Y$, i.e., $S'$ contains exactly $k$ elements from $X$. A $k$-uniform $S$-capturing hypergraph $H = H(X,S,k)$ has a vertex set $X$ and a hyperedge set consisting of all $k$-element subsets of $X$ captured by $S$. In case when $k=2$ and $S$ is convex these graphs are planar graphs, known as convex distance function Delaunay graphs. In this paper we prove that for any $k\geq 2$, any $X$, and any convex compact set $S$, the number of hyperedges in $H(X,S,k)$ is at most $(2k-1)|X| - k^2 + 1 - \sum_{i=1}^{k-1}a_i$, where $a_i$ is the number of $i$-element subsets of $X$ that can be separated from the rest of $X$ with a straight line. In particular, this bound is independent of $S$ and indeed the bound is tight for all "round" sets $S$ and point sets $X$ in general position with respect to $S$. This refines a general result of Buzaglo, Pinchasi and Rote stating that every pseudodisc topological hypergraph with vertex set $X$ has $O(k^2|X|)$ hyperedges of size $k$ or less.Comment: new version with a tight result and shorter proo

Separation by Convex Pseudo-Circles

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