Multiplicity of Centroidal Hyperplane Sections

Definitions and mathematical statement

Let $n\geq 1$ be an integer. A convex body is a compact convex set $K\subset{\mathbb{R}}^n$ with nonempty interior. Writing $|K|$ for its $n$-dimensional Lebesgue measure, define its centroid by \[ c(K)=\frac{1}{|K|}\int_K x\,{\,\mathrm{d}} x. \] For an affine hyperplane $H\subset{\mathbb{R}}^n$ containing $c(K)$, define the centroid of the section using the induced $(n-1)$-dimensional measure: \[ c_H(K\cap H)=\frac{\displaystyle\int_{K\cap H}x\,{\,\mathrm{d}}\mathcal H^{n-1}(x)}{\mathcal H^{n-1}(K\cap H)}. \] Here $\mathcal H^{n-1}$ denotes Hausdorff measure with its usual Euclidean normalization; when $n=1$, $\mathcal H^0$ is counting measure. Since $c(K)$ lies in the interior of $K$, the denominator is positive. Set \[ \mathcal S(K)=\left\{H\subset{\mathbb{R}}^n: H\text{ is an affine hyperplane},\ c(K)\in H,\ c_H(K\cap H)=c(K)\right\}, \qquad \mu(n)=\min_{\substack{K\subset{\mathbb{R}}^n\\K\text{ a convex body}}}\#\mathcal S(K). \] Distinct sections are counted by distinct affine hyperplanes, without orientation; no finiteness of $\mathcal S(K)$ is assumed.

The original question asks whether, for every convex body $K\subset{\mathbb{R}}^n$, \[ \#\mathcal S(K)\geq n+1, \] and, more generally, asks for the value of $\mu(n)$.

The supplied literature assessment, dated $2026\text{-}08\text{-}22$, reports \[ \mu(2)=3,\qquad \mu(n)=1\quad(n\geq5), \] and identifies $\mu(3)$ and $\mu(4)$ as unresolved. These are supplied dated curation claims, not independently verified mathematical results in this draft. The universal $n+1$ question is retained as the original question; the supplied assessment would rule out that bound in dimensions $n\geq5$. The record does not explicitly state a lower bound on the dimension; the definitions above use the standard positive-integer convention, including the zero-dimensional section convention when $n=1$.

Short English statement

Count the hyperplane sections through a convex body's centroid whose own centroids equal that point, and determine the smallest possible count in each dimension. The supplied dated assessment identifies dimensions $3$ and $4$ as the remaining open cases; its literature claims require independent review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006138$, title: Multiplicity of Centroidal Hyperplane Sections; exact source problem code: catalogue entry. Collection: Oberwolfach Reports Open Problems. The record gives proposed year $2024$ and names no proposer.
• Supplied source citation: Convex Geometry and its Applications (2025), Oberwolfach Reports, pp. 3301--3376, DOI 10.4171/owr/2024/58. https://doi.org/10.4171/owr/2024/58. The citation's year $2025$ and the record's proposed year $2024$ are preserved as supplied.
• Supplied literature reference: Myroshnychenko--Tatarko--Yaskin, Adv. Math. 2025. https://doi.org/10.1016/j.aim.2024.110081.
• Supplied literature reference: Haddad--Jimenez--Villa, 2026. https://doi.org/10.1007/s12220-026-02570-2.
• Literature assessment date: 2026-08-22. The supplied dated curation labels the problem partially solved and reports $\mu(2)=3$, $\mu(n)=1$ for $n\geq5$, and unresolved values in dimensions $3$ and $4$. This assessment is not independently verified here. The background calls this the Loewner-centroid minimum, without defining that terminology separately.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official Oberwolfach report. This is supplied provenance, not a source-verification claim by this draft.