Convex Body Uniqueness from Tangential Hyperplane Sections

Definitions and mathematical statement

A convex body in ${\mathbb{R}}^n$ means a compact convex set with nonempty interior. For the usual nondegenerate interpretation of the question, let $n\geq 2$ be an integer and let $t>0$. These parameter ranges are implicit conventions here; the supplied statement does not explicitly specify them. Let $c\in{\mathbb{R}}^n$, and define \[ B(c,t)=\{x\in{\mathbb{R}}^n:\|x-c\|\leq t\},\qquad S(c,t)=\partial B(c,t). \] Suppose that $K,L\subset{\mathbb{R}}^n$ are convex bodies and \[ S(c,t)\subset\operatorname{int}(K)\cap\operatorname{int}(L). \] By convexity, this also implies $B(c,t)\subset\operatorname{int}(K)\cap\operatorname{int}(L)$, so interpreting the enclosed sphere as its solid ball gives the same containment condition. For each unit vector $u\in{\mathbb{R}}^n$, let \[ H(u)=\{x\in{\mathbb{R}}^n:\langle x-c,u\rangle=t\}. \] These are precisely the hyperplanes tangent to the common sphere $S(c,t)$. Write $\operatorname{vol}_{n-1}$ for Euclidean $(n-1)$-dimensional volume in each such hyperplane. The question is whether, for every choice of the parameters and bodies above, \[ \left[\operatorname{vol}_{n-1}(K\cap H(u))=\operatorname{vol}_{n-1}(L\cap H(u))\text{ for every }u\in{\mathbb{R}}^n\text{ with }\|u\|=1\right] \quad\Longrightarrow\quad K=L. \] The source asks about sections by all tangent hyperplanes to one common sphere, with that sphere strictly inside both bodies. The supplied dated assessment identifies this as the Barker--Larman unique-determination question and labels it partially solved: it reports that the general convex-body case remains open even in dimension two, that Yaskin proved a positive result for convex polytopes, and that later work gives further local positive results. Those are supplied literature qualifications, not independently verified assertions here; precise hypotheses for the cited positive results are not supplied.

Short English statement

If two convex bodies contain the same sphere strictly inside them, and their sections by every hyperplane tangent to that sphere have equal $(n-1)$-dimensional volumes, must the bodies coincide? The draft uses the conventional ranges $n\geq2$ and $t>0$, which the source leaves implicit. The supplied partially solved label and reported special cases require independent review.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30001383$, source code catalogue entry, title Convex Body Uniqueness from Tangential Hyperplane Sections, collection Oberwolfach Reports Open Problems. The record lists proposed year 2009 and no individual proposer. Its attribution to Barker--Larman appears in the supplied literature assessment.
• Supplied report citation: Convex Geometry and its Applications (2010), Oberwolfach Reports, pp. 2851--2926, DOI 10.4171/owr/2009/53. This is also the literature source labeled "original Oberwolfach report 4136": https://doi.org/10.4171/owr/2009/53.
• Supplied literature source labeled "non-central sections open problem": https://arxiv.org/abs/1509.08174.
• Supplied literature source labeled "Yaskin polytope theorem": https://doi.org/10.1007/s00208-010-0538-y.
• Literature assessment date: 2026-08-21. The partially solved classification, reported openness for general convex bodies even in dimension two, reported polytope result, and unspecified later local positive results are supplied dated curation derived from the OWR record, not independently verified here. The source explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified; its supplied verification explanation says that the complete statement was preserved while mathematical notation was placed in delimiters. This records an editorial check and does not establish mathematical correctness or current literature status.