We investigate the support of a capacity-achieving input to a vector-valued Gaussian noise channel. The input is subjected to a radial even-moment constraint and is either allowed to take any value in ${\mathbb{R}}^n$ or is restricted to a given compact subset of ${\mathbb{R}}^n$. It is shown that the support of the capacity-achieving distribution is composed of a countable union of submanifolds, each with a dimension of $n-1$ or less. When the input is restricted to a compact subset of ${\mathbb{R}}^n$, this union is finite. Finally, the support of the capacity-achieving distribution is shown to have Lebesgue measure 0 and to be nowhere dense in ${\mathbb{R}}^n$.