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 $$\left( {\begin{array}{c}n 0\end{array}}\right) +\left( {\begin{array}{c}n 1\end{array}}\right) +\left( {\begin{array}{c}n 2\end{array}}\right) +\left( {\begin{array}{c}n 3\end{array}}\right) $$ . 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. For a fixed $$k \in \{1,\dots ,n-1\}$$ , we also count the number of elements in a maximal family of k-subsets of S separable by convex pseudo-circles. This time the number depends on the configuration of S, but it is again equal to the number of k-subsets of S separable by true circles. Thus, it is an invariant of S: it does not depend on the choice of the maximal family. To achieve these results, we introduce a graph that generalizes the dual graph of the order-k Voronoi diagram. The vertices of the graph are the elements of a maximal family of k-subsets of S separable by convex pseudo-circles. In order to count the number of these vertices, we show that the graph is realizable as a triangulation.