Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations

The Alexandrov-Fenchel inequality, a fundamental result in convex
geometry, has recently been shown to be one component within a broader
'Kahler package'. This structure was observed to emerge in different
areas of mathematics, including geometry, algebra, and combinatorics,
and encompasses Poincare duality, the hard Lefschetz theorem, and the
Hodge-Riemann relations. After unpacking these statements within the
context of this talk, I will explain where complex geometry intersects
with convex geometry in the proofs.

Based on joint work with Andreas Bernig and Jan Kotrbaty.