Geometric-Comparison Stability Framework

Definitions and mathematical statement

For notation, let $n$ be a positive integer and let \[ S^{n-1}=\{\xi\in{\mathbb{R}}^n:\|\xi\|_2=1\}. \] A convex body in ${\mathbb{R}}^n$ is a compact convex set with nonempty interior. It is origin-symmetric if $K=-K$. Write $|K|$ for its $n$-dimensional Lebesgue volume. The source does not specify the intended range of dimensions or any additional restrictions on the bodies.

The symbols $f_K$ and $f_L$ stand for unspecified geometric characteristics evaluated at directions $\xi\in S^{n-1}$. Their definitions and normalization are absent. Consequently, the following is only the supplied stability framework, not a fully specified mathematical conjecture.

For the intended characteristics and admissible origin-symmetric convex bodies $K,L\subset{\mathbb{R}}^n$, the source asks whether there is an absolute constant $c$ such that, for every $\varepsilon>0$, \[ \left[\forall\xi\in S^{n-1},\quad f_K(\xi)\leq f_L(\xi)+\varepsilon\right] \quad\Longrightarrow\quad |K|^{(n-1)/n}\leq |L|^{(n-1)/n}+c\varepsilon. \] The intended absolute constant would be independent of $K$, $L$, $\varepsilon$, and the admissible dimension $n$. Whether any dependence on the choice of characteristic is permitted cannot be determined from the record. The unspecified characteristics are not being universally quantified over all possible assignments $K\mapsto f_K$.

The supplied statement-verification note describes this inequality as a framework preceding applications to specified characteristics, and reports that those applications supply results. It therefore marks the standalone statement as unrecoverable and supplies no clean replacement. The registry nevertheless retains an open status label; that label is supplied metadata, not a verified assertion that this unspecified framework is an open problem.

Short English statement

The record asks whether a uniform directional comparison of two unspecified geometric characteristics yields a volume comparison with error proportional to $\varepsilon$. The characteristics, their normalization, and the precise admissible body class are missing, so a determinate standalone problem cannot be recovered.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002251$, title Geometric-Comparison Stability Framework, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No named proposer is supplied; the proposed-year field is $2012$.
• Supplied report citation: Convex Geometry and its Applications (2013), Oberwolfach Reports, pp. 3473--3548, DOI 10.4171/owr/2012/59. https://doi.org/10.4171/owr/2012/59. The record describes this collection as extracted from workshop reports and open-problem discussions.
• Supplied literature source labeled "Oberwolfach report": https://ems.press/content/serial-article-files/46430.
• Supplied literature source labeled "volume comparison stability paper": https://arxiv.org/abs/1207.6347. No authors or paper title are supplied in the record.
• Literature assessment dated 2026-08-21: the supplied curation labels the record open while stating that the omitted characteristic definitions and body-class hypotheses prevent identification of the exact assertion. It further states that general volume-comparison stability depends on those missing hypotheses. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The supplied verification note separately characterizes the displayed inequality as a framework whose reported applications supply results. These source descriptions and time-sensitive claims require independent verification.