On the boundedness of an iteration involving points on the hypersphere

For a finite set of points X on the unit hypersphere indwe consider the iteration ui+1 = ui+ χi, where χiis the point of X farthest from ui. Restricting to the case where the origin is contained in the convex hull of X we study the maximal length of ui. We give sharp upper bounds for the length of uiindependently of X. Precisely, this upper bound is infinity for d ≥ 3 and $\sqrt{2}$ for d = 2. © 2012 World Scientific Publishing Company.