Regular simplices and Gaussian samples

We show that if a suitable type of simplex in R ⁿ is randomly rotated and its vertices projected onto a fixed subspace, they are as a point set affine-equivalent to a Gaussian sample in that subspace. Consequently, affine-invariant statistics behave the same for both mechanisms. In particular, the facet behavior for the convex hull is the same, as observed by Affentranger and Schneider; other results of theirs are translated into new results for the convex hulls of Gaussian samples. We show conversely that the conditions on the vertices of the simplex are necessary for this equivalence. Similar results hold for random orthogonal transformations.