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.