Minkowski Problem for Asymptotic Convex Sets

Definitions and mathematical statement

Let $n\in{\mathbb{N}}$, $n\geq 3$. The supplied record uses a set $C$, a spherical domain $\Omega\subseteq S^{n-1}$, and a class of $C$-asymptotic convex sets $K\subseteq C$. It does not supply the hypotheses on $C$, the definition of $\Omega$ in terms of $C$, or the definition of $C$-asymptotic. Accordingly, let $\mathcal A_C$ denote that source-defined class, pending recovery of these definitions. For $K\in\mathcal A_C$, write $S_K$ for its surface-area measure on $\Omega$, with the convention intended in the source. The record does not explicitly specify that convention for these potentially unbounded convex sets.

For $0<\alpha<\pi/2$, put \[ \omega_\alpha=\{v\in\Omega:\operatorname{dist}(v,\partial\Omega)>\alpha\}. \] The distance and the boundary are those intended by the source; their metric and ambient-space conventions are not specified in the supplied record. All integrals below are nonnegative integrals over the open interval $(0,\pi/2)$, so no endpoint values of $\omega_\alpha$ are required.

The higher-dimensional characterization problem is to give necessary and sufficient conditions on a Borel measure $\mu$ on $\Omega$ for \[ \exists K\in\mathcal A_C\quad \forall\text{ Borel }E\subseteq\Omega,\qquad \mu(E)=S_K(E). \] The record does not state whether additional finiteness or regularity requirements are imposed on the candidate Borel measures. This formulation must therefore be read subject to the unresolved source definitions and measure conventions above.

The specific conjecture supplied with this problem is the necessary condition \[ \forall K\in\mathcal A_C,\qquad \int_0^{\pi/2}\left(S_K(\omega_\alpha)\right)^{1/(n-1)}\,{\,\mathrm{d}}\alpha<\infty. \] Thus the requested task includes proving or disproving this assertion for dimensions greater than two. The source does not assert that this integral condition alone characterizes the realizable measures.

The following contextual assertions are preserved as supplied source claims, not as independently verified results. The excerpt says that higher dimensions admit necessary conditions that are not sufficient and sufficient conditions that are not necessary, without specifying those conditions. It attributes to Schneider, reference [8] in the report, the estimate \[ S_K(\omega_\alpha)\leq \frac{c}{\alpha^{n-1}} \qquad(0<\alpha<\pi/2) \] for every $C$-asymptotic set $K$, with a finite constant $c$ independent of $\alpha$; its permitted dependence on the other data is not specified.

For dimension $2$, the excerpt attributes to Semenov--Zhao, reference [9], the following characterization: a Borel measure $\mu$ on $\Omega$ is the surface-area measure on $\Omega$ of a unique $C$-asymptotic set $K\subseteq C$ if and only if \[ \int_0^{\pi/2}\mu(\omega_\alpha)\,{\,\mathrm{d}}\alpha<\infty. \] The source describes this condition as slightly stronger than Schneider's necessary condition. The same unresolved setup conventions apply to this reported theorem.

In dimension $3$, the excerpt further attributes to Semenov--Zhao constructions showing that every larger exponent fails as a universal necessary condition: for each $p>1/2$, there exists a $C$-asymptotic set $K$ for which \[ \int_0^{\pi/2}\left(S_K(\omega_\alpha)\right)^p\,{\,\mathrm{d}}\alpha=\infty. \] The supplied wording does not clarify how $C$ is quantified in this construction. It also reports that, in dimension $3$, no condition of this integral form with a single power can be both necessary and sufficient. The excerpt says that the dimension-three computations can be adapted to general dimensions, with more complicated computations and notation; it does not supply the resulting precise higher-dimensional statements. Its suggestion that global growth conditions provide only part of the desired characterization is presented as the speakers' expectation.

Short English statement

For dimensions $n\geq3$, characterize which Borel measures on $\Omega$ occur as surface-area measures of $C$-asymptotic convex sets. In particular, determine whether every such set satisfies the displayed integral condition with exponent $1/(n-1)$. The supplied record reports a complete characterization in dimension $2$, but omits definitions and hypotheses needed for a fully self-contained formulation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006139$, title Minkowski Problem for Asymptotic Convex Sets; exact source problem code catalogue entry. The record lists proposed year 2024 and identifies no individual proposer.
• Supplied report citation: Convex Geometry and its Applications (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/58, pp. 3301--3376. https://doi.org/10.4171/owr/2024/58. The supplied title year is 2025, whereas the report identifier and proposed year use 2024; this discrepancy has not been resolved.
• Supplied literature source: Semenov--Zhao, 2024 preprint, https://arxiv.org/abs/2409.18699. The excerpt identifies Semenov--Zhao as its reference [9] and attributes the planar characterization and the dimension-three constructions to that work; no complete bibliographic entry is supplied.
• Supplied literature source: OWR 2024/58, https://ems.press/content/serial-article-files/50771.
• Schneider, reference [8] in the supplied report excerpt, is credited with the surface-area growth estimate. The record supplies no title, year, or complete bibliographic entry for [8].
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved, reports a complete necessary-and-sufficient characterization in dimension 2, and describes the higher-dimensional characterization as incomplete. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels its statement corrected_verified and says it was reconstructed and checked against the official Oberwolfach report. These are supplied verification claims; no source visit or independent verification is asserted here.