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Let R be the local ring of a point on a variety X over an algebraicallyclosed field k. We make a connection between the notion of mixed (Samuel)multiplicity of m-primary ideals in R and intersection theory of subspaces ofrational functions on X which deals with the number of solutions of systems ofequations. From this we readily deduce several properties of mixedmultiplicities. In particular, we prove a (reverse) Alexandrov-Fenchelinequality for mixed multiplicities due to Teissier and Rees-Sharp. As anapplication in convex geometry we obtain a proof of a (reverse)Alexandrov-Fenchel inequality for covolumes of convex bodies inscribed in aconvex cone.

On mixed multiplicities of ideals