  The algebra of smooth translation-invariant valuations on convex bodies,
introduced by S.Alesker in the early 2000s, was in part proved and in part
conjectured to satisfy properties formally analogous to those of the cohomology
ring of a compact K\"ahler manifold: Poincar\'e duality, the hard Lefschetz
theorem, and the Hodge-Riemann relations. Our main result establishes the hard
Lefschetz theorem and the Hodge-Riemann relations in full generality. As a
consequence, we obtain McMullen's quadratic inequalities, which are valid for
strongly isomorphic polytopes and known to fail in general, for convex bodies
with smooth and strictly positively curved boundary. Our proof is based on
elliptic operator theory and on perturbation theory applied to unbounded
operators on a natural Hilbert space completion of the space of smooth
translation-invariant valuations.
