The space of finitely additive measures on compact convex bodies (also called convex valuations) is an infinite-dimensional vector space that shares many properties with the cohomology algebra of a compact Kaehler manifold. In particular it is a graded algebra that satisfies a version of Poincaré duality. In a recent work with Jan Kotrbatý (Prague) and Thomas Wannerer (Jena) we prove a version of the mixed hard Lefschetz theorem and mixed Hodge-Riemann relations. The latter can be translated into new higher-order versions of the famous Alexandrov-Fenchel inequality for mixed volumes. The proof combines techniques from differential geometry and functional analysis.
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Dustin Clausen
