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 $u_{i+1}=u_i+\chi_i$, 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 $d=2$.

💡 Research Summary

The paper studies a simple geometric iteration defined on a finite set X of points lying on the unit hypersphere S^{d‑1}⊂ℝ^d. Starting from the zero vector, one repeatedly adds the point of X that is farthest from the current vector:

 u₀ = 0, u_{i+1} = u_i + χ_i, χ_i ∈ argmax_{x∈X}‖x – u_i‖.

Because several points may be equally far, the iteration is not unique; the authors consider the set U(X) of all vectors that can appear in any such run and define

 u*(X) = sup{‖u‖ : u ∈ U(X)}.

The motivation comes from the analysis of the Bădoi‑Clarkson algorithm for computing the smallest enclosing ball (SEB) of a point set. In that algorithm the error vector e_i = i(c_i – c) (c is the true SEB centre, c_i the i‑th approximation) satisfies exactly the same recurrence as u_i after a suitable scaling. Consequently, a finite bound on u*(X) translates into a faster convergence rate for the SEB algorithm.

The authors restrict attention to the case where the origin lies in the convex hull of X. They introduce a geometric classification called b‑balanced:

* b = 0 means the origin is outside conv X;
* 0 < b < d means the origin lies on a b‑dimensional face of conv X but not on any lower‑dimensional face;
* b = d means the origin is an interior point (the set is “balanced”).

The main results are:

1. Theorem 3: If X is 0‑balanced, then u*(X) = ∞. The proof is elementary: the point of X closest to the origin has distance ε > 0, so each step adds a vector of length at least ε, forcing linear growth of ‖u_i‖.

2. Theorem 4 (dimension d = 2):

   • u*_{2,0} = ∞ (as above).
   • u*{2,1} = u*{2,2} = √2.
     In other words, for any planar set X whose convex hull contains the origin (either on an edge or in its interior), the length of the iterates never exceeds √2, and this bound is attained by the simple set X = {(0,1), (1,0)}. The proof is intricate: the authors parametrize X by a “base‑gap” angle φ ∈