Limit Laws for Random-Simplex Volume Statistics

Definitions and mathematical statement

For each integer $d\geq 1$, let $K_d\subset{\mathbb{R}}^d$ be a convex body, meaning a compact convex set with nonempty interior. Let $X_{d,0},\ldots,X_{d,d}$ be independent random points with distribution $\operatorname{Vol}_d(K_d)^{-1}\operatorname{Vol}_d\!\restriction_{K_d}$, where $\operatorname{Vol}_d$ denotes $d$-dimensional Lebesgue measure. Define \[ P_{d+1}^{K_d}=\operatorname{conv}\{X_{d,0},\ldots,X_{d,d}\},\qquad V_d=\operatorname{Vol}_d(P_{d+1}^{K_d}). \] The first requested problem is to formulate and prove central limit theorems for $V_d$ as $d\to\infty$, separately when $K_d$ is a Euclidean ball, a regular simplex, a cube, or a cross-polytope. In particular, a central limit theorem for the volume itself would require appropriate deterministic centering constants $a_d\in{\mathbb{R}}$ and scaling constants $b_d>0$ such that \[ \frac{V_d-a_d}{b_d}\xrightarrow[d\to\infty]{\mathrm{distribution}}\mathcal N(0,1). \] The source does not specify these constants or the geometric normalization of the bodies. This displayed limit expresses the requested type of result, not an established conclusion. A central limit theorem for $\log V_d$ is a different statement and does not by itself supply the requested volume limit theorem.

Separately, for integers $d\geq4$ and $n\geq1$, let $U_1,\ldots,U_n$ be independent random variables uniformly distributed on the vertex set $\{0,1\}^d$, and define \[ P_n^d=\operatorname{conv}\{U_1,\ldots,U_n\}. \] Thus sampling is with replacement, as required by independence. The second requested problem is to determine the asymptotic behavior of \[ \mathbb E\left[f_\ell(P_n^d)\right],\qquad 2\leq\ell\leq d-2. \] The supplied record does not explicitly define $f_\ell$. Under the usual face-number interpretation, $f_\ell(P)$ counts the $\ell$-dimensional faces of $P$; this interpretation is conditional, because the supplied literature assessment instead describes intermediate intrinsic volumes. Face numbers and intrinsic volumes are distinct statistics. The record also does not specify which of $n,d,\ell$ tend to infinity, their relative growth, or the desired uniformity of an asymptotic formula. Consequently, the second request is not a fully specified asymptotic proposition.

The original statement additionally requests central limit theorems for unspecified "listed volume and face-volume statistics." No list or definitions of those additional statistics are supplied, so this portion cannot be formulated precisely from the record.

Short English statement

Seek volume central limit theorems for random simplices formed by independent uniform points in four families of convex bodies. Separately, seek asymptotics for $\mathbb E f_\ell(P_n^d)$ for convex hulls of independent uniform cube vertices. The meaning of $f_\ell$, the asymptotic regime, and the additional face-volume statistics require clarification. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003399$, source problem code catalogue entry, title: Limit Laws for Random-Simplex Volume Statistics. Collection: Oberwolfach Reports Open Problems. No individual proposer is identified in the supplied record.
• Supplied source citation: Mini-Workshop: Perspectives in High-Dimensional Probability and Convexity (2018), Oberwolfach Reports, pp. 499--526, DOI 10.4171/owr/2017/10. https://doi.org/10.4171/owr/2017/10. The record separately gives proposed year 2017; the differing year indications are retained without resolution.
• Supplied literature source, "Oberwolfach report": https://ems.press/content/serial-article-files/46672.
• Supplied literature source, "Alonso-Gutiérrez et al.": https://arxiv.org/abs/1906.02471. No fuller bibliographic details are supplied.
• Supplied literature source, "Reitzner": https://doi.org/10.1016/j.aim.2004.03.006. No fuller bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved, states that central limit theorems for log-volumes and several special models are known, and describes the full collection of volume central limit theorems and intermediate intrinsic-volume asymptotics as unresolved. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels its statement corrected_verified and reports comparison with the official Oberwolfach report. That verification claim is supplied metadata; no independent source verification is asserted here.