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