Topics in Geometry: Convexities

As convexity of the unit ball characterizes norms among translation-invariant homogeneous metrics on vector spaces, it is natural to study the implications of enhanced convexity assumptions. Those early "isometric" concepts are treated, but the course mainly focuses on deep "isomorphic" theories of enhanced convexity. The term "isomorphic" refers to the fact that the requirements are preserved under large perturbations. A feature of such concepts is the roundabout role that probability plays in their formulation and investigation, as they entail understanding the behavior of vector-valued independent random variables and martingales.