On the boundedness of an iteration involving points on the hypersphere

For a finite set of points $X$ on the unit hypersphere in $\mathbb{R}^d$ we consider the iteration $u_{i+1}=u_i+χ_i$, where $χ_i$ 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 $d=2$.

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