In recent years, mathematicians have developed new approaches to study convex sets: instead of considering convex sets themselves, they explore certain functions or measures that are related to them. Problems from convex geometry become thereby accessible to analytic and probabilistic tools, and we can use these tools to make progress on very difficult open problems.

We discuss in this Snapshot such a functional ex- tension of some “volumes” which measure how “big” a set is. We recall the construction of “intrinsic vol- umes”, discuss the fundamental inequalities between them, and explain the functional extensions of these results.