Ellipsoid Rigidity from Constant Central-Section Products

Definitions and mathematical statement

Let $n\in{\mathbb{Z}}$ with $n\geq 3$. Equip ${\mathbb{R}}^n$ with its Euclidean inner product $\langle\cdot,\cdot\rangle$, and let $S^{n-1}=\{\theta\in{\mathbb{R}}^n:\langle\theta,\theta\rangle=1\}$. A convex body is a compact convex subset of ${\mathbb{R}}^n$ with nonempty interior. It is origin-symmetric if $K=-K$. Define its support function by \[ h_K(\theta)=\max_{x\in K}\langle x,\theta\rangle, \] and define $\theta^\perp=\{x\in{\mathbb{R}}^n:\langle x,\theta\rangle=0\}$. Write $\operatorname{vol}_{n-1}$ for the Euclidean $(n-1)$-dimensional volume on this hyperplane. An origin-centered ellipsoid is a set of the form \[ \{x\in{\mathbb{R}}^n:x^{\mathsf T}Ax\leq 1\}, \] where $A$ is a real symmetric positive-definite $n\times n$ matrix.

Question. For every integer $n\geq 3$ and every origin-symmetric convex body $K\subseteq{\mathbb{R}}^n$, does \[ \exists c>0\ \ \forall\theta\in S^{n-1},\qquad h_K(\theta)\operatorname{vol}_{n-1}(K\cap\theta^\perp)=c \] imply that $K$ is an origin-centered ellipsoid? The constant $c$ may depend on $K$ and $n$, but must be independent of $\theta$. Positivity of $c$ follows from the convex-body and symmetry assumptions.

The supplied curation identifies this question with the global fifth Busemann--Petty assertion and labels it partially solved. It reports a result for bodies sufficiently close to the Euclidean ball, but does not specify the topology, quantitative closeness condition, or complete hypotheses of that result. This background qualification is not an additional hypothesis of the global question, and the reported literature assessment has not been independently verified.

Short English statement

If the product of the support function and the volume of the perpendicular central section is constant in every direction, must an origin-symmetric convex body in every dimension $n\geq 3$ be an ellipsoid? The supplied dated curation reports a result near the Euclidean ball and labels the global question partially solved; this draft does not independently verify that assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004030$, title: Ellipsoid Rigidity from Constant Central-Section Products. Collection: Oberwolfach Reports Open Problems. Exact source problem code: catalogue entry. The record gives proposed year 2018 and does not name a proposer.
• Supplied source citation: Convex Geometry and its Applications (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/54, pp. 3217--3281. https://doi.org/10.4171/owr/2018/54.
• Supplied literature reference: Alfonseca et al., Advances in Mathematics 2021. https://www.sciencedirect.com/science/article/pii/S0001870821003595.
• Supplied literature reference: BIRS report on Busemann--Petty problems. https://www.birs.ca/workshops/2020/20w5037/report20w5037.pdf.
• Literature assessment dated exactly 2026-08-22: the supplied curation states that the global fifth Busemann--Petty assertion is not known and that Alfonseca et al. prove it only for bodies sufficiently close to the Euclidean ball. This is supplied dated curation derived from the OWR record, not an independently verified assessment. The source itself requests independent verification of time-sensitive status and cited claims.