Invertible relations on polytopes

Ifg andh are any nonzero functions on the class of convex polytopes then (Fⁱ,F^j) =g(Fⁱ)/h(F^j) is a valuation whose inverse is (Fⁱ,F^j) = (–1)^j–ih(Fⁱ)/g(F^j). This is proved and a smaller class of valuations are characterized: those (Fⁱ,F^j) which depend only oni andj and which have inverses of the same form.