  Given an affine variety X and a finite dimensional vector space of regular
functions L on X, we associate a convex body to (X, L) such that its volume is
responsible for the number of solutions of a generic system of functions from
L. This is a far reaching generalization of usual theory of Newton polytopes
(which is concerned with toric varieties). As applications we give new, simple
and transparent proofs of some well-known theorems in both algebraic geometry
(e.g. Hodge Index Theorem) and convex geometry (e.g. Alexandrov-Fenchel
inequality). Our main tools are classical Hilbert theory on degree of
subvarieties of a projective space (in algebraic geometry) and Brunn-Minkowski
inequality (in convex geometric).
