Extremizers of Bézout Mixed-Volume Invariants

Definitions and mathematical statement

The supplied record uses the invariants $b(K)$ and $b_2(K)$ without defining either of them. Their formulas, domains of admissible test bodies, normalization, and relationship are unavailable in the supplied evidence. Consequently, the following is a conditional formulation of the recorded targets, not a complete definition of the invariants.

Under the customary convention, a convex body in ${\mathbb{R}}^n$ is a compact convex subset with nonempty interior. An $n$-simplex is the convex hull of $n+1$ affinely independent points of ${\mathbb{R}}^n$. The record does not explicitly confirm the nonempty-interior convention or specify the permitted dimensions $n$; these qualifications require review.

For each intended dimension $n$, let $\mathcal K_n$ denote the intended class of convex bodies. With the missing definitions of $b$ and $b_2$ restored, the recorded questions are:

1. Characterize the class \[ \{K\in\mathcal K_n:b(K)=n\}. \]
2. Is an $n$-simplex a minimizer of $b_2$ on $\mathcal K_n$, and must every minimizer be an $n$-simplex? Here a minimizer means a body $K\in\mathcal K_n$ satisfying \[ b_2(K)\leq b_2(L)\qquad\text{for every }L\in\mathcal K_n. \] The record does not specify an equivalence relation for uniqueness, establish attainment of the minimum, or supply its value.

The quoted report passage states that the characterization of bodies satisfying $b_2(K)=2$ is known, whereas the characterization of bodies satisfying $b(K)=n$ remains open. No characterization or connection between this level set and the minimization question is supplied. The passage also announces a necessary condition on the support of the surface area measure of potential minimizers, but does not state that condition. These are supplied contextual assertions, not mathematical results established by this draft.

Short English statement

The record asks which convex bodies satisfy $b(K)=n$, and whether simplices are precisely the possible minimizers of $b_2$. Neither invariant is defined in the supplied material, so a fully specified mathematical statement requires the missing definitions and dimension conventions. The supplied partially solved status and literature claims remain subject to independent review.

Sources

• Local UnsolvedMath registry, permanent ID $30005727$, title: Extremizers of Bézout Mixed-Volume Invariants; source problem code catalogue entry. The record belongs to 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 the supplied authoritative source citation; its contents were not independently consulted for this draft.
• Supplied literature source: Extremizers in Soprunov--Zvavitch inequalities. https://doi.org/10.1016/j.jmaa.2023.127461.
• Supplied literature source: Soprunov--Zvavitch, Bezout inequality for mixed volumes. https://doi.org/10.1093/imrn/rnv390.
• Supplied literature source: original Oberwolfach report. https://doi.org/10.4171/owr/2023/51.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that the simplex characterization is proved for polytopes and in dimension three, while the general convex-body minimizer and $b(K)=n$ characterization remain open. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was checked against the official report. This verification is supplied registry metadata, not a source check performed for this draft.