Thin-Shell Conjecture for Schatten Balls

Definitions and mathematical statement

For a standard full-matrix interpretation, fix $\mathbb K\in\{{\mathbb{R}},{\mathbb{C}}\}$, let $n\geq 1$ be an integer, and regard $M_n(\mathbb K)$ as a real Euclidean space with inner product $\langle A,B\rangle=\operatorname{Re}\operatorname{tr}(A^*B)$. Its real dimension is $d_n=n^2$ for $\mathbb K={\mathbb{R}}$ and $d_n=2n^2$ for $\mathbb K={\mathbb{C}}$. If $s_1(A),\ldots,s_n(A)$ are the singular values of $A$, define, under the usual normed-space convention $1\leq p\leq\infty$, \[ \|A\|_{S_p}=\left(\sum_{j=1}^n s_j(A)^p\right)^{1/p}\quad(1\leq p<\infty),\qquad \|A\|_{S_\infty}=\max_{1\leq j\leq n}s_j(A), \] and $B_p^n=\{A:\|A\|_{S_p}\leq1\}$. Let $Y_{n,p}$ be uniformly distributed on $B_p^n$ with respect to real Lebesgue measure. This distribution is centered by symmetry. Write $\Sigma_{n,p}$ for its covariance operator and set $X_{n,p}=\Sigma_{n,p}^{-1/2}Y_{n,p}$. Thus $X_{n,p}$ is isotropic and ${\mathbb{E}}\|X_{n,p}\|_{\mathrm{HS}}^2=d_n$, where $\|A\|_{\mathrm{HS}}^2=\operatorname{tr}(A^*A)$.

A common squared-radius variance formulation of the requested thin-shell bound is the existence of an absolute constant $C<\infty$ such that \[ \operatorname{Var}\!\left(\|X_{n,p}\|_{\mathrm{HS}}^2\right) ={\mathbb{E}}\left(\|X_{n,p}\|_{\mathrm{HS}}^2-d_n\right)^2 \leq C d_n \] for all dimensions and parameters in the intended family. The supplied question asks whether the thin-shell conjecture holds for the Schatten balls at the remaining values of $p$ outside the known large-$p$ results.

Unresolved qualifications. The record does not specify those remaining values or dimension-dependent ranges, the scalar field, or whether full matrices or a restricted matrix space are intended. Consequently, the definitions above give a conditional standard interpretation, rather than a uniquely recovered source formulation. The record also does not state the exact thin-shell normalization or whether a bound uniform over all remaining $p$ is requested. A fixed-$p$ version would instead ask, for each remaining $p$, for a constant $C_p$ independent of $n$. These missing choices prevent a fully determined quantified statement.

Short English statement

Do the Schatten unit balls satisfy the thin-shell conjecture throughout the parameter ranges left outside the known large-$p$ results? In a standard isotropic variance formulation, this asks for a dimension-independent bound on the variance of squared Euclidean radius divided by dimension. The remaining parameter ranges and several matrix-space and uniformity conventions are unspecified in the supplied record.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30003067$, source code catalogue entry, title Thin-Shell Conjecture for Schatten Balls; collection: Oberwolfach Reports Open Problems. The supplied record gives proposed year 2016 and names no individual proposer.
• Supplied report citation: Asymptotic Geometric Analysis (2016), Oberwolfach Reports, DOI 10.4171/owr/2016/11, pp. 507--565. https://doi.org/10.4171/owr/2016/11. The background describes these reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or attributed speaker is supplied.
• Supplied literature source, labeled "thin-shell and slicing progress": https://arxiv.org/abs/2303.14938. The record supplies no authors, title, or precise supporting result for this link.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open, reports control of thin-shell variance in large-$p$ ranges, and states that the report leaves the remaining $p$ as an explicit question and that no general resolution of those ranges was found. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review. These are supplied editorial labels; they do not establish the missing mathematical conventions or independently verify the open status.