Maximum-Volume Sections of $ell_p$ Balls

Definitions and mathematical statement

For an integer $n\geq 2$ and a real number $2<p<\infty$, define the unit ball \[ B_p^n=\left\{x\in{\mathbb{R}}^n:\sum_{j=1}^n |x_j|^p\leq 1\right\}. \] For a $k$-dimensional linear subspace $E\subseteq{\mathbb{R}}^n$, let $\operatorname{vol}_k$ denote the Euclidean $k$-dimensional volume induced on $E$. A linear section is $B_p^n\cap E$, so the section passes through the origin.

The supplied literature assessment identifies the target as maximum-volume hyperplane sections. Under this interpretation, the problem is: for every integer $n\geq 2$ and every $2<p<\infty$, determine all linear hyperplanes $H\subseteq{\mathbb{R}}^n$ such that \[ \operatorname{vol}_{n-1}(B_p^n\cap H) \geq \operatorname{vol}_{n-1}(B_p^n\cap L) \quad\text{for every linear hyperplane }L\subseteq{\mathbb{R}}^n. \] Equivalently, writing $S^{n-1}=\{a\in{\mathbb{R}}^n:\sum_{j=1}^n a_j^2=1\}$ and $a^\perp=\{x\in{\mathbb{R}}^n:\sum_{j=1}^n a_jx_j=0\}$, determine every $a\in S^{n-1}$ satisfying \[ \operatorname{vol}_{n-1}(B_p^n\cap a^\perp) = \max_{b\in S^{n-1}}\operatorname{vol}_{n-1}(B_p^n\cap b^\perp). \] The normals $a$ and $-a$ represent the same hyperplane.

Unresolved scope qualification. The supplied clean statement says only "linear sections" and does not specify their dimension or the ambient dimension. If it intends sections of arbitrary fixed dimension, the corresponding question is to determine, for each $1\leq k<n$, every $k$-dimensional linear subspace $E$ satisfying \[ \operatorname{vol}_k(B_p^n\cap E) \geq \operatorname{vol}_k(B_p^n\cap F) \quad\text{for every }F\subseteq{\mathbb{R}}^n\text{ with }\dim F=k. \] The supplied evidence does not establish that this broader scope is intended. Volumes are compared only within a fixed section dimension. The unit-radius convention fixes scale; a ball of radius $r>0$ multiplies each $k$-section volume by $r^k$ and leaves its maximizing subspaces unchanged.

The original excerpt also contains a corrupted moment inequality involving $q$, random variables $U_j$, and a symbol $p_n^*$. Their definitions and the missing expression are not supplied, so neither that inequality nor a precise phase-transition conjecture can be reconstructed here.

Short English statement

For each $2<p<\infty$, identify the central hyperplanes cutting the largest Euclidean volume from an $\ell_p$ unit ball in ${\mathbb{R}}^n$. This follows the supplied literature assessment; the clean statement leaves the section dimension unspecified. The record labels the problem open, subject to its supplied dated assessment and unresolved scope.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30004954$, source problem code catalogue entry, Oberwolfach Reports Open Problems collection. The supplied record lists proposed year $2021$ and gives no individual proposer.
• Supplied source citation: Convex Geometry and its Applications (2022), Oberwolfach Reports, pp. 3187--3266, DOI 10.4171/owr/2021/59. https://doi.org/10.4171/owr/2021/59. This is also the supplied literature-source link labelled "original OWR report." The citation year and the year embedded in the DOI are retained as supplied.
• Supplied literature-source label: "Konig on maximal lp sections." https://arxiv.org/abs/2409.06432. No fuller bibliographic details are supplied in the record.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem open and states that maximal hyperplane sections remain unknown in the range $2<p<\infty$, with recent primary work treating restricted regimes and supporting a proposed phase transition. This is supplied dated curation, not an independently verified assessment; it does not assert that every parameter case is unresolved.
• The supplied record marks its statement corrected_verified and reports checking it against the official Oberwolfach report. That is a source-record verification claim, not verification performed for this draft. The original excerpt mentions results of Latala and Oleszkiewicz as reference [9] and a further result as reference [2], but supplies neither bibliography entry nor enough intact notation to state those results precisely.