In 1987 Cheng determined $\phi_p$-optimal designs for linear regression (without intercept) over the $n$-dimensional unit cube $\lbrack 0, 1\rbrack^n$ for $-\infty \leq p \leq 1$. These are uniform distributions on the vertices with a fixed number of entries equal to unity, and mixtures of neighboring such designs. In 1989 Pukelsheim showed that this class of designs is essentially complete and that the corresponding class of moment matrices is minimally complete, with respect to what he called Kiefer ordering. In this paper, these results are extended to general permutation-invariant design regions.
"Complete Class Results for the Moment Matrices of Designs Over Permutation-Invariant Sets." Ann. Statist. 23 (1) 41 - 54, February, 1995. https://doi.org/10.1214/aos/1176324454