Approaching capacity by equiprobable signaling on the Gaussian channel

For the additive white Gaussian noise channel, there is a gap between the channel capacity and the highest achievable rate of equiprobable uniformly spaced one-dimensional signaling. It is commonly believed that approaching channel capacity requires the constituent one-dimensional signal points to have a Gaussian probability distribution. It is shown that the channel capacity can be achieved by equiprobable signaling with geometrical, Gaussian-like signal sets. Construction of these signal constellations is explicitly given. This result implicates that it is possible to approach channel capacity without using any kind of shaping codes