Algebraic $k$-sets and generally neighborly embeddings

TL;DR

This paper explores algebraic analogs of $k$-sets and $k$-facets, providing exact counts for certain algebraic curves and introducing generally neighborly embeddings to analyze point set configurations.

Contribution

It introduces algebraic $k$-sets and $k$-facets, derives exact counts for specific algebraic curves, and studies generally neighborly embeddings to understand point set arrangements.

Findings

Exact count of halving conic sections for certain point sets

Introduction of generally neighborly embeddings

Improved bounds on $k$-sets/$k$-facets in convex position

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.