Special Seminar in Applied Mathematics

While intrinsic volumes of convex bodies (compact convex sets in euclidean space) are long-studied quantities with a rich and well-established theory, the study of intrinsic volumes of spherical convex bodies, or equivalently closed convex cones, is still in its infancy with many fascinating open questions. On the other hand, it has already shown that this theory may be a powerful tool for answering certain questions arising for example in the domain of convex programming. The goal of this lecture series is to provide an access to this theory, to present some recent successful applications, and to highlight the most prominent open questions.

Part 2: What is the probability that the solution of a random semidefinite program has rank r?

The second lecture is devoted to the intrinsic volumes of symmetric cones, which include the cone of (real symmetric) positive semidefinite matrices. We present closed formulas for these intrinsic volumes and use them to provide an answer to the question in the title of this talk (which has been asked already 15 years ago). This answer is achieved by a neat application of the kinematic formulas, which we will discuss in greater depth.