Assuming in a d-dimensional space, we have samples from n class. The best way of separating samples of each class from each other is to have the samples from each class as far as possible from every other class.
If d = n-1, then the solution is obviously a simplex  where observations of each class are centered in the corners of a d-dimensional simplex, which is a linearly separable space (similar to LDA space).
But how about when d >= n? Is there a solution to find n coordinates in a d-dimensional space, where each point is in an equal distance from every other point?