Busemann-Petty Problem for Arbitrary Measures

Definitions and mathematical statement

The supplied record concerns a Busemann--Petty comparison for arbitrary measures, but its retained passage refers to an omitted "above question." Its remaining comparison is \[ \mu(K)\leq\mu(L). \] For this expression to be meaningful, $K$ and $L$ must be measurable for a common measure $\mu$. This is a requirement for interpreting the notation, not a recovery of the omitted hypotheses. The record does not specify the ambient space, the admissible sets $K,L$, the admissible measures $\mu$, or the comparison assumptions from which the displayed inequality is supposed to follow. In particular, no section-comparison hypothesis is supplied.

The recoverable question is therefore only: under the omitted hypotheses of the source's preceding question, must $\mu(K)\leq\mu(L)$? A precise quantified mathematical proposition cannot be recovered from the supplied passage alone.

The supplied passage reports that the answer is independent of the "choice" of measure and depends only on the dimension $n$, being affirmative for $n\leq4$ and negative for $n\geq5$. The scope of the measures and geometric hypotheses in this reported dichotomy remains unspecified. This is a supplied account of an established result, not a theorem verified here or a newly formulated open problem.

Short English statement

The intended comparison asks whether $\mu(K)\leq\mu(L)$ follows from missing assumptions. The supplied account reports an affirmative answer for $n\leq4$ and a negative answer for $n\geq5$, but the extracted text does not determine a precise mathematical statement.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30006129$, source problem code catalogue entry, title "Busemann-Petty Problem for Arbitrary Measures." No individual proposer is identified; the supplied proposed year is 2024.
• Supplied report citation: Convex Geometry and its Applications (2025), Oberwolfach Reports, pp. 3301--3376, DOI 10.4171/owr/2024/58. https://doi.org/10.4171/owr/2024/58. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. The retained passage attributes the dimensional dichotomy to references [6, 7], whose bibliographic identities are not supplied.
• Supplied literature reference: Zvavitch, Math. Ann. 2005. https://doi.org/10.1007/s00208-004-0611-5. The record does not explicitly identify this reference with either [6] or [7].
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem "partially solved" and reports that the arbitrary-measure Busemann--Petty comparison is affirmative for $n\leq4$ and negative for $n\geq5$, so there is no dimension-independent answer. This assessment is supplied dated curation derived from the OWR material, not independently verified here; its status and cited claims require independent verification.
• The supplied statement-verification field says that the official report explicitly describes this arbitrary-measure version as settled with the same dimensional dichotomy as the classical problem. The supplied statement-status field is unrecoverable. These are registry assertions, not independent verification of the report.