Capacity-Achieving Input Distributions of Additive Vector Gaussian Noise Channels: Even-Moment Constraints and Unbounded or Compact Support

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 [Formula: see text] or is restricted to a given compact subset of [Formula: see text]. It is shown that the support of the capacity-achieving distribution is composed of a countable union of submanifolds, each with a dimension of [Formula: see text] or less. When the input is restricted to a compact subset of [Formula: see text] , 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 [Formula: see text].