  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.
