Asymptotic Volume Parameters for Euclidean Balls

Definitions and mathematical statement

The supplied record asks two questions about Euclidean balls, denoted by $B_2^n$, as the positive integer dimension $n$ tends to infinity. It does not define the functional $\Theta$, the sets $\widetilde E_A$, their dependence on $n$, or the measure denoted by $\operatorname{vol}$. The normalization of the balls and of volume is also unspecified. Consequently, the questions below retain undefined source notation and require the missing definitions before they constitute a fully specified mathematical problem.

1. Is the sequence $\left(\Theta(B_2^n)\right)_{n\geq 1}$ bounded? If its entries are real scalars, the boundedness assertion is \[ \exists C\in[0,\infty)\ \forall n\in{\mathbb{Z}}_{\geq 1},\qquad |\Theta(B_2^n)|\leq C. \] The scalar interpretation is conditional because the record does not specify the definition or codomain of $\Theta$.
2. Is it true that, for every fixed real number $A>0$, \[ \operatorname{vol}(\widetilde E_A)=1-o(1)\qquad(n\to\infty)? \] Here the dependence of $\widetilde E_A$ and possibly of the volume convention on $n$ is implicit in the supplied notation. The displayed assertion asks that the volume converge to $1$ for each fixed $A>0$; it does not assert uniform convergence in $A$. No one-sided estimate follows from the notation $1-o(1)$ alone.

Short English statement

The source asks whether the quantities $\Theta(B_2^n)$ remain bounded across dimensions and whether, for every fixed $A>0$, the volume of $\widetilde E_A$ tends to $1$. Definitions of both quantities and their normalizations are missing, so this is an incomplete statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30001386$, title Asymptotic Volume Parameters for Euclidean Balls, exact source problem code catalogue entry, collection Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2009.
• Supplied report citation: Convex Geometry and its Applications (2010), Oberwolfach Reports, DOI 10.4171/owr/2009/53, pp. 2851--2926. Supplied source URL and literature-source link labeled "original OWR report": https://doi.org/10.4171/owr/2009/53. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The report contents have not been independently consulted for this draft.
• Supplied literature assessment, checked exactly 2026-08-21: status "open." The supplied assessment says that the OWR report explicitly poses both Euclidean-ball questions as challenging open problems and that a targeted literature check found no later primary resolution. This is supplied dated curation, not an independently verified assessment of the report or current literature; the record requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified. Its accompanying verification states only that TeX notation was enclosed in math delimiters without changing the mathematical content; this does not establish the missing definitions or mathematical correctness.