Geometric-Comparison Separation Framework

Definitions and mathematical statement

For a positive integer $n$, write $S^{n-1}=\{\xi\in{\mathbb{R}}^n:\|\xi\|_2=1\}$. A convex body is a compact convex subset of ${\mathbb{R}}^n$ with nonempty interior; it is origin-symmetric if $K=-K$. Write $|K|$ for its $n$-dimensional Lebesgue volume.

The source associates symbols $f_K,f_L$ with origin-symmetric convex bodies $K,L\subset{\mathbb{R}}^n$, calling them geometric characteristics, but does not define these characteristics or specify any additional restrictions on the bodies. Their values at directions $\xi\in S^{n-1}$ occur in the following comparison.

The supplied separation question has the conditional template: does there exist an absolute constant $c>0$ such that, for every admissible dimension $n$, every admissible pair $K,L$, and every $\varepsilon>0$, \[ \left[\forall\xi\in S^{n-1},\quad f_K(\xi)\leq f_L(\xi)-\varepsilon\right] \quad\Longrightarrow\quad |K|^{(n-1)/n}\leq |L|^{(n-1)/n}-c\varepsilon? \] Here "absolute" is understood as independence from $n,K,L,\varepsilon$. The admissible dimensions, any additional body hypotheses, the assignment $K\mapsto f_K$, and any allowed dependence of $c$ on a fixed choice of characteristic are unresolved. The condition is imposed separately for each positive $\varepsilon$ satisfying the antecedent.

This is a framework transcription, not a recovered determinate conjecture. The supplied verification labels the statement unrecoverable and describes it as a companion framework to proved applications in the contribution; it says that the extraction does not preserve a determinate open claim.

Short English statement

The supplied question asks whether a uniform positive directional gap between unspecified geometric characteristics forces a proportional gap between the indicated powers of the bodies' volumes. Missing definitions and hypotheses prevent a precise conjecture. The registry's open label is supplied curation, not an independently verified mathematical status.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30002252$, title "Geometric-Comparison Separation Framework," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2012 and names no proposer.
• 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 supplied background describes these reports as documenting workshops and their 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.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the item open while stating that the companion separation inequality omits geometric definitions and the body class, so no definitive status is available without those hypotheses. The background calls this dated literature triage and requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification field says that the contribution presents a generic separation framework accompanying proved applications, that no determinate open claim survives the extraction, and that no separate clean statement is promoted. These source-verification assertions have not been independently checked here.