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This paper concerns the facial geometry of the set of n×n correlation matrices. The main result states that almost every set of r vertices generates a simplicial face, provided that r≤cn, where c is an absolute constant. This bound is qualitatively sharp because the set of correlation matrices has no simplicial face generated by more than 2n vertices. [ABSTRACT FROM AUTHOR]

The intrinsic volumes of a convex cone are geometric functionals that return basic structural information about the cone. Recent research has demonstrated that conic intrinsic volumes are valuable for understanding the behavior of random convex optimization problems. This paper develops a systematic technique for studying conic intrinsic volumes using methods from probability. At the heart of this approach is a general Steiner formula for cones. This result converts questions about the intrinsic volumes into questions about the projection of a Gaussian random vector onto the cone, which can then be resolved using tools from Gaussian analysis. The approach leads to new identities and bounds for the intrinsic volumes of a cone, including a near-optimal concentration inequality. [ABSTRACT FROM AUTHOR]