Banach–Mazur Diameter of Nonsymmetric Convex Bodies

Definitions and mathematical statement

Let $n\geq 1$ be an integer, and let $\mathcal K_n$ denote the class of compact convex subsets of ${\mathbb{R}}^n$ with nonempty interior. Under the standard affine interpretation of the supplied formula, define \[ d_{\mathrm{BM}}(K,L)=\inf\left\{\lambda\geq 1:\ T_1(K)\subseteq T_2(L)\subseteq \lambda T_1(K),\quad T_1,T_2:{\mathbb{R}}^n\to{\mathbb{R}}^n\text{ invertible affine maps}\right\}. \] Here $\lambda T_1(K)=\{\lambda x:x\in T_1(K)\}$. The affine maps allow translations as well as invertible linear changes of coordinates. The affine Banach--Mazur compactum consists of the affine equivalence classes of these bodies; the distance above is expressed in its multiplicative convention.

Determine, as a function of $n$, the exact value of \[ D_n=\sup_{K,L\in\mathcal K_n}d_{\mathrm{BM}}(K,L), \] called the diameter in the supplied record, which describes this extremal value as a maximum.

There are unresolved source qualifications. The excerpt does not explicitly specify the admissible maps $T_1,T_2$; their invertible affine interpretation is supported by its description of comparison up to affine transformations. The formulation above includes all convex bodies without imposing central symmetry. The source's phrase "nonsymmetric convex bodies" may instead mean bodies that are strictly not centrally symmetric. Under that reading the domain would be \[ \mathcal K_n^{\mathrm{ns}}=\{K\in\mathcal K_n:\text{there is no }c\in{\mathbb{R}}^n\text{ with }K=2c-K\}, \] and the intended extremum and its attainment require clarification; in particular, this class is empty for $n=1$. The supplied record does not resolve these readings.

The background excerpt also discusses centrally symmetric bodies and attributes an ellipsoid comparison to John's theorem, labeled [3]. Its displayed dilation factor is corrupted and the bibliographic entry [3] is absent, so that comparison is not adopted as a precise additional assertion here.

Short English statement

For each dimension, determine the largest multiplicative Banach--Mazur distance between convex bodies when affine transformations are allowed. The intended meaning of "nonsymmetric" and the incomplete source formula require review. The supplied status is partially solved; this draft does not independently verify that assessment.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30006507$, source problem code catalogue entry; Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is 2025.
• Supplied report citation: Functional Inequalities: Geometric Calculus meets Stochastic Analysis (2026), Oberwolfach Reports, pp. 2917--2994, DOI 10.4171/owr/2025/54. https://doi.org/10.4171/owr/2025/54. The record also labels this source Oberwolfach Report 2025/54.
• Supplied literature source: Bizeul, Distances between non-symmetric convex bodies. https://arxiv.org/abs/2510.20511.
• Literature assessment checked at 2026-08-22: the supplied dated curation labels the problem partially solved and states that the exact diameter remains unknown, while available bounds are optimal up to polylogarithmic factors. This assessment and the cited claims have not been independently verified here.
• The registry labels the statement corrected_verified and says that only formula delimiters, notation, grammar, and typography were repaired. This is supplied verification metadata, not an independent mathematical review of this draft.