Let $X_1, \cdots, X_n \sim \mathrm{Unif}[0,1]$ be $n$ random variables, each with marginal distribution being a standard uniform distribution. I want to characterize the set of covariance matrices (or correlation matrix if this is easier) that can be attained by these $n$ variables. Is there a simple characterization? I feel the set should be a polyhedron.