Classical Busemann-Petty Problem

Definitions and mathematical statement

The supplied record is titled "Classical Busemann--Petty Problem," but its statement contains only the question \[ \text{Does it follow that }\operatorname{Vol}_n(K)\leq\operatorname{Vol}_n(L)\text{?} \] Here $\operatorname{Vol}_n$ conventionally denotes $n$-dimensional Euclidean volume when applied to suitable measurable subsets of ${\mathbb{R}}^n$. This explains the notation only: the excerpt does not specify the admissible dimensions, the domains or properties of $K,L$, or the hypotheses from which the displayed inequality is supposed to follow. The accompanying presentation title mentions convex bodies, but does not supply these missing hypotheses. Consequently, neither a complete implication nor its quantifiers can be recovered from the supplied statement.

The record's verification field identifies the passage as a recounting of the classical Busemann--Petty question. Its supplied literature assessment reports an affirmative answer in dimensions at most four and counterexamples in dimensions at least five. These are supplied contextual claims, not independently verified results in this draft, and they do not recover the omitted antecedent. The record labels the statement unrecoverable and its literature status partially_solved; no precise unresolved mathematical target is specified.

Short English statement

The excerpt asks whether $\operatorname{Vol}_n(K)\leq\operatorname{Vol}_n(L)$ follows from hypotheses that are missing. Although the record identifies the classical Busemann--Petty problem and supplies a dimension-dependent literature assessment, a complete quantified statement cannot be recovered without adding unsupported assumptions.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30006128$, source code catalogue entry, title "Classical Busemann--Petty Problem," in the Oberwolfach Reports Open Problems collection.
• 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 record separately gives proposed year $2024$; the relationship between these year labels is not explained.
• Attribution accompanying the extracted question: "Measure comparison problems for dilations of convex bodies," Artem Zvavitch, joint work with Malak Lafi. The record does not identify a proposer of this particular question.
• Supplied literature reference: Gardner--Koldobsky--Schlumprecht, Annals 1999. https://annals.math.princeton.edu/articles/12537.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the status "partially solved" and reports an affirmative answer for dimensions at most four and counterexamples for dimensions at least five. This is supplied dated curation derived from the OWR record, not independently verified here. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification field says the official report recounts the already answered classical Busemann--Petty question. This is a supplied verification claim; neither linked source has been independently consulted for this draft.