Algebraic k-Sets and Generally Neighborly Embeddings

Brett Leroux, Luis Rademacher

Abstract: Given a set S of n points in \({{\mathbb {R}}}^d\), a k-set is a subset of k points of S that can be strictly separated by a hyperplane from the remaining \(n-k\) points. Similarly, one may consider k-facets, which are hyperplanes that pass through d points of S and have k points on one side. A notorious open problem is to determine the asymptotics of the maximum number of k-sets. In this paper we study a variation on the k-set/k-facet problem with hyperplanes replaced by algebraic surfaces. In stark contrast to the original k-set/k-facet problem, there are some natural families of algebraic curves for which the number of k-facets can be counted exactly. For example, we show that the number of halving conic sections for any set of \(2n+5\) points in general position in the plane is \(2\left( {\begin{array}{c}n+2\\ 2\end{array}}\right) ^2\). To understand the limits of our argument we study a class of maps we call generally neighborly embeddings, which map generic point sets into neighborly position. Additionally, we give a simple argument which improves the best known bound on the number of k-sets/k-facets for point sets in convex position.