On the boundedness of an iteration involving points on the hypersphere

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 $\sqrt{2}$ for $d=2$.