A protrusive ordering of 5 points not witnessed by any finite multiset

TL;DR

This paper constructs a specific set of five points in the plane with a protrusive ordering that cannot be replicated by any finite multiset distance ranking, answering a question in combinatorial geometry.

Contribution

It provides a counterexample demonstrating that not all protrusive orderings can be derived from finite multiset distance sums.

Findings

Existence of a 5-point set with a protrusive ordering not obtainable by sum-of-distances ranking

Answers an open question by Alon, Defant, Kravitz, and Zhu

Highlights limitations of distance-based ordering methods in geometry

Abstract

Given a finite set of points $C \subseteq \mathbb{R}^d$, we say that an ordering of $C$ is protrusive if every point lies outside the convex hull of the points preceding it. We give an example of a set $C$ of $5$ points in the Euclidean plane possessing a protrusive ordering that cannot be obtained by ranking the points of $C$ according to the sum of their distances to a finite multiset of points. This answers a question of Alon, Defant, Kravitz and Zhu.