High-Dimensional Volumes of Random Simplices

Definitions and mathematical statement

The supplied record asks for high-dimensional asymptotics of expected random-simplex volumes. Its sampling law, dimension convention, body scaling, and normalization are not specified, so it does not determine a unique sequence of expectations.

A parameterized formulation makes these missing data explicit. For each positive integer $d$, let $m_d\geq 1$ be an ambient dimension, let $K_d\subset{\mathbb{R}}^{m_d}$ be a compact convex set with nonempty interior, and let $\mu_d$ be a Borel probability measure on $K_d^{m_d+1}$. Let $(X_{d,0},\ldots,X_{d,m_d})$ have joint law $\mu_d$, and define \[ S_d=\operatorname{conv}\{X_{d,0},\ldots,X_{d,m_d}\},\qquad V_d=\operatorname{vol}_{m_d}(S_d), \] where $\operatorname{vol}_{m_d}$ is $m_d$-dimensional Lebesgue measure; this volume is zero when the vertices are affinely dependent. The expected volume is \[ A_d(K_d,\mu_d)={\mathbb{E}}[V_d] =\int_{K_d^{m_d+1}} \operatorname{vol}_{m_d}\!\left(\operatorname{conv}\{x_0,\ldots,x_{m_d}\}\right) \,{\,\mathrm{d}}\mu_d(x_0,\ldots,x_{m_d}). \] This framework does not assume that the vertices are independent or uniformly distributed: neither assumption is supplied by the record.

The requested task is to determine the asymptotic behavior of these expected volumes as $d\to\infty$ for the intended sequences of regular simplices, cubes, and other standard convex bodies, together with the intended normalized asymptotic quantities. The supplied literature assessment also explicitly mentions cross-polytopes. This is a formulation template requiring specification of $m_d$, $K_d$, $\mu_d$, and the normalization before a precise asymptotic target can be stated.

In particular, the source calls the random object a "$(d+1)$-simplex." In the usual dimensional terminology this has dimension $d+1$ and has $d+2$ vertices. A convex hull of $d+1$ points is instead at most $d$-dimensional. The record does not specify the ambient dimension or resolve whether its wording refers to dimension or to the number of sampled vertices. No identification of $m_d$ with $d$ or $d+1$ is therefore imposed. The scale of each containing body and the meaning of "normalized quantities" also remain unresolved.

Short English statement

Determine the high-dimensional asymptotics of the expected volume of a random simplex in a regular simplex, a cube, and other intended standard convex bodies, including the requested normalized quantities. The record leaves the dimension convention, vertex distribution, body scaling, and normalization unspecified; these must be resolved for a unique mathematical problem. Its supplied status is partially solved, which is a dated curation label rather than an independently verified conclusion.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30003398$, source problem code catalogue entry, title High-Dimensional Volumes of Random Simplices, in Oberwolfach Reports Open Problems. The record gives proposed year 2017 and names no proposer.
• Supplied source citation: Mini-Workshop: Perspectives in High-Dimensional Probability and Convexity (2018), Oberwolfach Reports, pp. 499--526, DOI 10.4171/owr/2017/10. Supplied link: https://doi.org/10.4171/owr/2017/10. The background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature source, labeled "Oberwolfach report": https://ems.press/content/serial-article-files/46672.
• Supplied literature source, labeled "Alonso-Gutiérrez et al.": https://arxiv.org/abs/1906.02471. The record supplies no further bibliographic details for this reference.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that high-dimensional central-limit results and moment formulas cover important random-simplex models, while the requested asymptotics for the regular simplex, cube, and cross-polytope are not all known in the stated form. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; the source itself requests independent verification of time-sensitive status and cited claims.
• The registry supplies the statement-status label corrected_verified and reports a manual semantic, typographic, and LaTeX review claiming self-containment and preservation of the target. That supplied review does not resolve the formulation gaps identified here. No linked source was visited or independently checked for this draft.