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 1: Introduction and a small application

The first talk is an elementary introduction to the theory of intrinsic volumes. We describe both the classical euclidean as well as the spherical case, and show how the intrinsic volumes may (at least theoretically) be computed for polyhedral and for smooth cones. We also introduce the kinematic formulas and give an answer to the following question: Asymptotically, does every point in an n-dimensional unit sphere lie in an ε-neighborhood of a cn-dimensional subsphere (ε > 0 and c ∈ [0, 1] fixed)?