On the boundedness of an iteration involving points on the hypersphere

TL;DR

This paper investigates the boundedness of an iterative process involving points on a hypersphere, establishing sharp upper bounds for the length of the iterates depending on the dimension, with the bound being infinite in dimensions three and higher.

Contribution

The paper provides the first sharp upper bounds for the maximal length of the iteration on the hypersphere, showing unboundedness in higher dimensions and a specific bound in two dimensions.

Findings

Boundedness depends on the dimension of the hypersphere.

The maximal length is infinite for dimensions three and higher.

The maximal length is exactly or dimension two.

Abstract

For a finite set of points $X$ on the unit hypersphere in $\mathbb{R}^d$ we consider the iteration $u_{i+1}=u_i+\chi_i$, where $\chi_i$ is the point of $X$ farthest from $u_i$. Restricting to the case where the origin is contained in the convex hull of $X$ we study the maximal length of $u_i$. We give sharp upper bounds for the length of $u_i$ independently of $X$. Precisely, this upper bound is infinity for $d\ge 3$ and $\sqrt2$ for $d=2$.