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

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.