Simplicial faces of the set of correlation matrices

This paper concerns the facial geometry of the set of $n\times n$ correlation matrices. The main result states that almost every set of $r$ vertices generates a simplicial face, provided that $r \leq \sqrt{$\mathrm{c}$ n}$,where$$\mathrm{c}$$ is an absolute constant. This bound is qualitatively sharp because the set of correlation matrices has no simplicial face generated by more than $\sqrt{2n}$ vertices.