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 vector-valued Gaussian noise channel. The is subjected radial even-moment constraint and either allowed take any value in Rn or restricted given compact subset Rn. It shown that distribution composed countable union submanifolds, each with dimension n−1 less. When Rn, this finite. Finally, have Lebesgue measure 0 be nowhere dense