  Statements analogous to the Hard Lefschetz Theorem (HLT) and the
Hodge-Riemann bilinear relations (HRR) hold in a variety of contexts: they
impose restrictions on the cohomology algebra of a smooth compact K\"ahler
manifold or on the intersection cohomology of a projective toric variety; they
restrict the local monodromy of a polarized variation of Hodge structure; they
impose conditions on the possible $f$-vectors of convex polytopes. While the
statements of these theorems depend on the choice of a K\"ahler class, or its
analog, there is usually a cone of possible K\"ahler classes. It is then
natural to ask whether the HLT and HRR remain true in a mixed context. In this
note we present a unified approach to proving the mixed HLT and HRR,
generalizing the previously known results, and proving it in new cases such as
the intersection cohomology of non-rational polytopes.
