Mixed volumes and mixed integrals

Abstract

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 extension of some “volumes” which measure how “big” a set is. We recall the construction of “intrinsic volumes”, discuss the fundamental inequalities between them, and explain the functional extensions of these results.