Hodge–Riemann Relations for Convex Valuations

Definitions and mathematical statement

Let $n$ denote the ambient dimension implicit in the supplied record; its allowed range is not stated. Under the usual interpretation, $\operatorname{Val}^{\infty}$ denotes smooth, continuous, translation-invariant valuations on compact convex subsets of an $n$-dimensional real vector space. A valuation satisfies \[ \phi(A\cup B)+\phi(A\cap B)=\phi(A)+\phi(B) \] whenever the sets involved are compact and convex. Smoothness conventionally refers to smoothness of the orbit map for the natural general linear group action. The coefficient field and precise algebra structure are not specified in the record.

Write $V(K_1,\ldots,K_n)$ for mixed volume. For a body $C$ in the source's class $\mathcal K_+^{\infty}$, define \[ \psi_C(K)=V(C,\underbrace{K,\ldots,K}_{n-1\text{ copies}}), \qquad \mathcal K=\{\psi_C:C\in\mathcal K_+^{\infty}\}. \] Here $\mathcal K_+^{\infty}$ is described by the source record as the class of smooth strictly convex bodies. Whether this notation additionally requires everywhere positive curvature is unresolved in the supplied material. The displayed valuation is homogeneous of degree $n-1$.

Question. Do the hard Lefschetz theorem and the Hodge--Riemann bilinear relations hold in $\operatorname{Val}^{\infty}$ with respect to every $\psi_C\in\mathcal K$?

Unresolved formulation details. The supplied record does not define the algebra operation, the Lefschetz operator, its degree conventions, the primitive subspaces, or the bilinear or Hermitian form and its sign convention. Consequently, the named relations cannot be expanded into fully specified isomorphism and positivity assertions from this record alone. In particular, choosing the product or convolution interpretation would require an additional qualification. The question above preserves the supplied target without making that choice.

The supplied context states that the conjecture is known for the one-dimensional cone consisting of $\psi_D$ associated with Euclidean balls $D$. Its dated curation describes the full smooth-convex-body cone as unresolved and labels the problem partially solved. These are supplied contextual assessments, not independently verified conclusions of this draft.

Short English statement

The question asks whether every mixed-volume valuation associated with a smooth strictly convex body satisfies the hard Lefschetz and Hodge--Riemann properties in the algebra of smooth valuations. The supplied context reports the Euclidean-ball cone case as established. Precise algebraic conventions and the exact meaning of the body class remain to be checked.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30005728$, title Hodge--Riemann Relations for Convex Valuations, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2023.
• Supplied report citation: Mini-Workshop: Positivity and Inequalities in Convex and Complex Geometry (2024), Oberwolfach Reports, pp. 2903--2940, DOI 10.4171/owr/2023/51. https://doi.org/10.4171/owr/2023/51. This is also the supplied literature-source link, labeled "original Oberwolfach report."
• Literature assessment dated 2026-08-22: the supplied curation labels the problem partially solved, reports the Euclidean-ball cone case as proved, and reports the full cone as unresolved. This is supplied dated curation, not independently verified literature assessment.
• The registry labels the statement corrected_verified and says it was checked against the official report. That verification claim is supplied metadata; this draft has not independently consulted the report.