Growing-Parameter Limits for Convex-Body Functionals

Definitions and mathematical statement

Let $j\in{\mathbb{N}}$, and let $(d_j)_{j\geq1}$ and $(k_j)_{j\geq1}$ be sequences of integers such that, for every $j$, \[ 0\leq k_j\leq d_j-1,\qquad 2k_j>d_j+1, \qquad\text{and}\qquad d_j\longrightarrow\infty\quad(j\to\infty). \] The source denotes by $Y^*_{d,k}$ a family of standardized random variables. The supplied record does not define the underlying random variables, their probability laws, or the standardization. Thus $Y^*_{d,k}$ must remain an unresolved source-dependent symbol; these omissions prevent a self-contained mathematical statement.

Problem, conditional on supplying that definition. Determine the asymptotic behavior of $Y^*_{d_j,k_j}$ along admissible sequences. Identify conditions on $(d_j,k_j)$ under which there exists a limiting probability distribution $\mu$ with \[ Y^*_{d_j,k_j}\xrightarrow{\mathrm{d}}\mu \qquad(j\to\infty), \] and identify the possible limiting distributions, including whether they are Gaussian. In particular, investigate the additional condition \[ \frac{k_j}{d_j}\longrightarrow\frac12. \] This condition is proposed as a regime to investigate, not asserted to imply convergence.

The supplied excerpt says that the characteristic function can be calculated explicitly, while the supplied background reports divergent cumulants. Neither the characteristic-function formula nor the relevant cumulant orders, parameter regimes, or divergence statements are provided, so these remarks do not supply the missing definition or additional precise hypotheses.

Short English statement

For the source's standardized random variables $Y^*_{d,k}$, study distributional limits as $d\to\infty$ with $0\leq k\leq d-1$ and $2k>d+1$, particularly when $k/d\to1/2$, and determine whether any limits are Gaussian. The family and its standardization are undefined in the supplied record, so this is a conditional, statement-only draft requiring mathematical review.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30006134$, title "Growing-Parameter Limits for Convex-Body Functionals," exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems.
• Supplied report citation: Convex Geometry and its Applications (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/58, pp. 3301--3376. https://doi.org/10.4171/owr/2024/58. The record gives proposed year 2024 and supplies no named proposer.
• Supplied literature source: "OWR 2024/58, open problem." https://ems.press/content/serial-article-files/50771.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, describes the limiting distribution in the growing-parameter regime as unresolved, and reports divergent cumulants. This is supplied dated curation, not 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 says it was reconstructed and checked against the official Oberwolfach report. These are supplied verification claims, not independent verification performed for this draft.