  This paper studies the capacity of an $n$-dimensional vector Gaussian noise
channel subject to the constraint that an input must lie in the ball of radius
$R$ centered at the origin. It is known that in this setting the optimizing
input distribution is supported on a finite number of concentric spheres.
However, the number, the positions and the probabilities of the spheres are
generally unknown. This paper characterizes necessary and sufficient conditions
on the constraint $R$ such that the input distribution supported on a single
sphere is optimal. The maximum $\bar{R}_n$, such that using only a single
sphere is optimal, is shown to be a solution of an integral equation. Moreover,
it is shown that $\bar{R}_n$ scales as $\sqrt{n}$ and the exact limit of
$\frac{\bar{R}_n}{\sqrt{n}}$ is found.
