Convex bodies and multiplicities of ideals

We associate convex regions in $\mathbb R^n$ to $\mathfrak m$-primary graded sequences of subspaces, in particular $\mathfrak m$-primary graded sequences of ideals, in a large class of local algebras (including analytically irreducible local domains). These convex regions encode information about Samuel multiplicities. This is in the spirit of the theory of Gröbner bases and Newton polyhedra on the one hand, and the theory of Newton–Okounkov bodies for linear systems on the other hand. We use this to give a new proof as well as a generalization of a Brunn–Minkowski inequality for multiplicities due to Teissier and Rees–Sharp.