Newton-Okounkov bodies and Segre classes,

Paolo Aluffi

Given a homogeneous ideal in a polynomial ring over **C**, we
adapt the construction of Newton-Okounkov bodies to obtain a convex
subset of Euclidean space such that a suitable integral over this set
computes the Segre zeta function of the ideal. That is, we extract the
numerical information of the Segre class of a subscheme of projective
space from an associated (unbounded) Newton-Okounkov convex set. The
result generalizes to arbitrary subschemes of projective space the
numerical form of a previously known result for monomial schemes.