Fixed-Delta Convolution-Body Extremizers

Definitions and mathematical statement

For a fixed ambient dimension $n$, a convex body in ${\mathbb{R}}^n$ means a compact convex set with nonempty interior, and $|\cdot|$ denotes $n$-dimensional Lebesgue volume. Fix $\delta\in(0,1)$ and a prescribed volume $V>0$, and set \[ \mathcal K_{n,V}=\{K\subseteq{\mathbb{R}}^n:K\text{ is a convex body and }|K|=V\}. \] The supplied corrected statement asks for a characterization of \[ \mathcal M_{\delta,n,V} = \left\{K\in\mathcal K_{n,V}: |C_\delta K|\geq |C_\delta L| \text{ for every }L\in\mathcal K_{n,V} \right\}. \] Here $C_\delta K$ denotes the associated convolution body named in the source record. Its construction, normalization, and boundary convention are not supplied, so the displayed formulation is conditional on specifying that operator. The intended ambient dimension is also unspecified; the display does not assert that the question was posed in every dimension. No existence of a maximizer is assumed.

The field labelled original statement contains a damaged extraction mixing Question 10 with Theorem 8. It mentions a symmetric planar radial set with radial function $\rho(v)=\cos^2(2mv)$, $v\in[0,2\pi]$, and an assertion involving every $\delta\in(0,1)$, some $m\in{\mathbb{N}}$, and a positive second derivative at $t=0$. The differentiated expression and the family depending on $t$ cannot be recovered reliably from the supplied text. This fragment supplies neither a usable definition of $C_\delta$ nor a reliable dimension restriction for the corrected extremizer question.

Short English statement

For fixed $\delta\in(0,1)$ and prescribed positive volume, characterize the convex bodies whose associated convolution bodies have maximal volume. The supplied record does not define the convolution-body operator or specify the intended ambient dimension, so a fully determined mathematical statement requires review of those points.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30006127$, title "Fixed-Delta Convolution-Body Extremizers," exact source code catalogue entry. No individual proposer is identified; the supplied proposed year is 2024.
• Supplied report citation: Convex Geometry and its Applications (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/58, pp. 3301--3376. Supplied source link: https://doi.org/10.4171/owr/2024/58.
• Supplied literature reference: "OWR 2024/58, Question 10," https://ems.press/content/serial-article-files/50771.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, identifies Question 10 as the fixed-$\delta$ extremizer problem, and reports that no authoritative later resolution of the full convex-body classification was found. This is supplied dated curation, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry labels its corrected statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied provenance claims, not verification performed for this draft. The damaged original extraction does not independently establish the corrected formulation.