Algebraic $k$-sets and generally neighborly embeddings

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 $2\binom{n+2}{2}^2$. To understand the limits of our argument we study a class of maps we call \emph{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.