Extending the centerpoint theorem to multiple points

TL;DR

This paper extends the centerpoint theorem to multiple points, proposing a set-based approach to higher-dimensional data representation that generalizes quantiles and relates to weak epsilon-nets and approximations.

Contribution

It introduces a novel extension of the centerpoint concept to multiple points, providing a higher-dimensional quantile-like framework.

Findings

Defines a set of points such that halfspaces containing one point include a large fraction of P

Establishes a relationship between this set and concepts of weak epsilon-nets and approximations

Proposes a method for selecting representative points in high-dimensional data

Abstract

The centerpoint theorem is a well-known and widely used result in discrete geometry. It states that for any point set $P$ of $n$ points in $\mathbb{R}^d$, there is a point $c$, not necessarily from $P$, such that each halfspace containing $c$ contains at least $\frac{n}{d+1}$ points of $P$. Such a point $c$ is called a centerpoint, and it can be viewed as a generalization of a median to higher dimensions. In other words, a centerpoint can be interpreted as a good representative for the point set $P$. But what if we allow more than one representative? For example in one-dimensional data sets, often certain quantiles are chosen as representatives instead of the median. We present a possible extension of the concept of quantiles to higher dimensions. The idea is to find a set $Q$ of (few) points such that every halfspace that contains one point of $Q$ contains a large fraction of the points of $P$ and every halfspace that contains more of $Q$ contains an even larger fraction of $P$. This setting is comparable to the well-studied concepts of weak $\varepsilon$-nets and weak $\varepsilon$-approximations, where it is stronger than the former but weaker than the latter.