  We prove that the multiplicity of a filtration of a local ring satisfies
various convexity properties. In particular, we show the multiplicity is convex
along geodesics. As a consequence, we prove that the volume of a valuation is
log convex on simplices of quasi-monomial valuations and give a new proof of a
theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In
another direction, we generalize a theorem of Rees on multiplicities of ideals
to filtrations and characterize when the Minkowski inequality for filtrations
is an equality under mild assumptions.
