Given A := {a1,...,am} C ℝn, we consider the problem of reducing the input set for the computation of the minimum enclosing ball of A. In this note, given an approximate solution to the minimum enclosing ball problem, we propose a simple procedure to identify and eliminate points in.A that are guaranteed to lie in the interior of the minimum-radius ball enclosing A. Our computational results reveal that incorporating this procedure into two recent algorithms proposed by Yildirim lead to significant speed-ups in running times especially for randomly generated largescale problems. We also illustrate that the extra overhead due to the elimination procedure remains at an acceptable level for spherical or almost spherical input sets.