Abstract. Generalizing the notion of Newton polyhedron, we define the Newton convex body, respectively, for semigroups of integral points, graded algebras, and linear series on varieties. We prove that any semigroup in the lattice Z n is asymptotically approximated by the semigroup of all the points in a sublattice and lying in a convex cone. Applying this we obtain several results: we show that for a wide class of graded algebras, the Hilbert functions have polynomial growth and their growth coefficients satisfy a Brunn-Minkowski type inequality. We prove analogues of Fujita approximation theorem for semigroups of integral points and graded algebras, which implies a generalization of it for arbitrary linear series. Applications to intersection theory include a far-reaching generalization of Kuˇsnirenko theorem (from Newton polyhedra theory) and a new version of Hodge inequality. We also give elementary proofs of Alexandrov-Fenchel inequality (and its corollaries) in convex geometry and their analogues in algebraic geometry. Key words: semigroups of integral points, convex body, mixed volume, Alexandrov
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