Monotonicity Constants for Expected Random-Simplex Volumes

Definitions and mathematical statement

Let $d\in{\mathbb{N}}$, $d\geq1$. A convex body in ${\mathbb{R}}^d$ means a compact convex set with nonempty interior. Write $V_d$ for $d$-dimensional Lebesgue measure. For each convex body $H\subset{\mathbb{R}}^d$, let $X_0^H,\ldots,X_d^H$ be independent random points with uniform distribution on $H$, meaning that \[ \Pr(X_i^H\in A)=\frac{V_d(A\cap H)}{V_d(H)} \] for every Borel set $A\subseteq{\mathbb{R}}^d$. Define the random simplex and its expected volume by \[ P_{d+1}^H=\operatorname{conv}\{X_0^H,\ldots,X_d^H\}, \qquad m_d(H)={\mathbb{E}}\left[V_d(P_{d+1}^H)\right]. \] Here $0<m_d(H)<\infty$. No joint distribution between the samples in different bodies is needed.

Does there exist a sequence of finite positive constants $(c(d))_{d\geq1}$, depending only on the dimension, such that \[ \forall d\geq1\ \forall K,L\subset{\mathbb{R}}^d\text{ convex bodies},\qquad K\subseteq L\ \Longrightarrow\ m_d(K)\leq c(d)m_d(L), \] and \[ \limsup_{d\to\infty}c(d)^{1/d}<\infty? \] Define the optimal dimensional constant, allowing the value $+\infty$, by \[ c_*(d)=\sup\left\{\frac{m_d(K)}{m_d(L)}: K,L\subset{\mathbb{R}}^d\text{ are convex bodies and }K\subseteq L\right\}. \] The source also asks for lower bounds for $c_*(d)$, without specifying a desired numerical or asymptotic strength. Its original wording calls this lower-bound task equivalent to the existence question; that equivalence is not justified by the supplied record, so both requests are retained separately.

The supplied dated curation labels the problem partially solved and reports that the monotonicity assertion corresponding to $c(d)=1$ is false in dimensions at least three. It further describes the quantitative question as connected to slicing-type bounds and unresolved at the requested sharp level. These are supplied literature assessments, not independently verified conclusions of this draft; the record does not specify what sharp level is intended.

Short English statement

For nested convex bodies, compare the expected volumes of simplices formed from independent uniform samples of $d+1$ points. Ask whether their ratio admits a bound depending only on dimension with at most exponential growth, and determine lower bounds for the optimal constant. The source's claimed equivalence between these requests and its intended sharpness remain unresolved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003397$, source problem code catalogue entry, titled "Monotonicity Constants for Expected Random-Simplex Volumes," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied report 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 describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, "Oberwolfach report": https://ems.press/content/serial-article-files/46672.
• Supplied literature source, "Kunis--Reichenwallner--Reitzner": https://doi.org/10.1007/s00454-017-9914-7.
• Supplied literature source, "Rademacher": https://doi.org/10.1112/S0025579311002063.
• Literature assessment checked on 2026-08-21, according to the supplied record. This is supplied dated curation derived from the OWR record, not independently verified source consultation. It labels the problem partially solved, reports failure of monotonicity in dimensions at least three, and describes the quantitative question as tied to slicing-type bounds and unresolved at the requested sharp level. Time-sensitive status and the cited claims require independent verification.