You need to know: Euclidean space 
, subspace of , subspace spanned by a set, cone, convex cone, closed cone, apex of a cone, groups, group 
of vectors with integer coordinates, subgroup, subgroup generated by a set, semigroup.
Background: Let 
be a semigroup. Let G be the subgroup of generated by S, L the subspace of spanned by S, and C the smallest closed convex cone (with apex at the origin) containing S.
The Theorem: On 21st April 2009, Kiumars Kaveh and Askold Khovanskii submitted to arxiv a paper in which they proved the following result. Let 
be a closed strongly convex cone that intersects the boundary (in the topology of the linear space L) of the cone C only at the origin. Then there exists a constant 
(depending on 
) such that any point in the group G that lies in and whose distance from the origin is bigger than N belongs to S.
Short context: Semigroup 
is called regularization of S. From the definition, S’ contains S. The Theorem states that the regularization S’ asymptotically approximates the semigroup S. The authors call this the approximation theorem. It is not very difficult to prove, but the authors showed that it has many applications in convex and algebraic geometry, intersection theory, and other areas of mathematics.
Links: Free arxiv version of the original paper is here, journal version is here.
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