Simplex Equality Cases in Convex Geometric Inequalities

Definitions and mathematical statement

Let $n\in{\mathbb{N}}$ with $n>2$, and let $K\subset{\mathbb{R}}^n$ be a convex body, meaning a compact convex set with nonempty interior. Write $|K|$ for its $n$-dimensional Lebesgue measure and $\operatorname{ext}(K)$ for its extreme points: $x\in K$ is extreme if $x=ty+(1-t)z$, with $y,z\in K$ and $0<t<1$, implies $y=z=x$. Define \[ g_K=\frac{1}{|K|}\int_K x\,{\,\mathrm{d}} x, \qquad C_2(K)=\frac{1}{|K|}\int_K\|x\|_2^2\,{\,\mathrm{d}} x. \] Let $r$ be the radius parameter appearing in the supplied statement, and assume \[ \|v\|_2\ge r\qquad\text{for every }v\in\operatorname{ext}(K). \] Unresolved parameter qualification. The supplied record does not explicitly specify the admissible range of $r$, in particular whether $r>0$ is required or $r=0$ is permitted. The conjecture below retains this unresolved qualification; no additional hypothesis on $r$ is silently imposed.

The supplied record describes the following bound as sharp: \[ C_2(K)\ge\frac{r^2+(n+1)\|g_K\|_2^2}{n+2}. \] Conjectured equality characterization. Under these hypotheses, for an arbitrary convex body $K$, \[ C_2(K)=\frac{r^2+(n+1)\|g_K\|_2^2}{n+2} \quad\Longleftrightarrow\quad \begin{gathered} K=\operatorname{conv}\{v_0,\ldots,v_n\},\\ v_0,\ldots,v_n\text{ are affinely independent},\\ \|v_i\|_2=r\quad(0\le i\le n). \end{gathered} \] Thus the sphere is centered at the origin. The original statement asks for this equivalence; the supplied expanded statement emphasizes the implication from equality to the simplex description. The original also refers to the hypotheses of an unreproduced Theorem 1, so completeness of the expanded hypotheses requires source review.

Short English statement

For $n>2$, consider a full-dimensional convex body whose extreme points have distance at least $r$ from the origin. The conjecture says that equality in the supplied second-moment bound holds exactly for simplices with all vertices on the sphere of radius $r$ centered at the origin. The admissible range of $r$ remains unspecified. The supplied dated assessment reports the equality characterization as established for polytopes and conjectural for general convex bodies.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001385$, title: Simplex Equality Cases in Convex Geometric Inequalities; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. The record gives proposed year 2009 and names no individual proposer.
• Supplied report citation: Convex Geometry and its Applications (2010), Oberwolfach Reports, pp. 2851--2926, DOI: https://doi.org/10.4171/owr/2009/53. This is also the supplied literature source labeled "original OWR report."
• Supplied literature source labeled "Fradelizi Paouris Schutt simplices in the Euclidean ball": https://doi.org/10.4153/CMB-2011-142-1. The supplied assessment identifies this as a 2012 primary paper proving the inequality and characterizing equality for polytopes, while explicitly leaving the equality characterization for general convex bodies as a conjecture.
• Literature assessment date: 2026-08-21. This is supplied dated curation derived from the OWR curation, not an independently verified assessment. The supplied status is "open." Time-sensitive status and cited mathematical claims require independent verification.
• The registry labels the statement corrected_verified and reports that the official OWR report supplies the moment functional, centroid term, sharp bound, and conjectured equality characterization for arbitrary convex bodies. These are supplied verification claims; no source visit or independent verification is asserted here.