In this talk, I will discuss some convexity properties of multiplicities of filtrations on a local ring. In particular, the multiplicity function is convex along geodesics. One corollary is that the volume function is strictly log convex on a simplex of quasi-monomial valuations, which answers a question of C. Li, X. Wang and Xu. As another major application, this gives a new proof of a theorem due to Xu and Zhuang on the uniqueness of normalized volume minimizers. In order to characterize strict convexity, we introduce the notion of saturation of a filtration, which turns out to be useful in other settings. For example, it allows us to generalize a theorem of Rees on multiplicities of ideals and characterize when the Minkowski inequality for filtrations is an equality.

This talk is based on joint work with Harold Blum and Yuchen Liu.