Intrinsic-Volume Deficits of Random Spherical Polytopes

Definitions and mathematical statement

Let $K$ be a spherically convex body contained in an open hemisphere of its ambient sphere. Here spherical convexity means that the shorter great-circle arc joining any two points of $K$ lies in $K$. The source does not explicitly specify the ambient sphere or the dimension denoted by $d$.

Let $X_1,\ldots,X_n$ be independent random points with the uniform distribution on $K$, and put \[ P_n=\operatorname{conv}_{\mathrm{sph}}\{X_1,\ldots,X_n\}, \] where $\operatorname{conv}_{\mathrm{sph}}$ denotes the smallest spherically convex set containing the indicated points. The usual interpretation of uniform sampling is normalized spherical volume: if $\mu$ is spherical volume on the ambient sphere and $0<\mu(K)<\infty$, then \[ \mathbb P(X_i\in A)=\frac{\mu(A\cap K)}{\mu(K)} \] for each measurable set $A$. This measure convention and the full-dimensionality needed for it require confirmation from the source.

The record refers to intrinsic-volume functionals $U_j$, but supplies neither their definitions nor their indexing and normalization conventions. Retaining these symbols without assigning them an unsupported definition, the proposed question is whether, for every body $K$ satisfying the intended hypotheses and every $j\in\{1,\ldots,d\}$, \[ \mathbb E\left[U_j(K)-U_j(P_n)\right] \asymp n^{-(d+1-j)}\qquad(n\to\infty). \] Here the two-sided interpretation of "of order" is that there exist constants $0<c_{K,j}\le C_{K,j}<\infty$ and an integer $n_0$, possibly depending on $K$, $j$, and the fixed ambient dimension, such that for every integer $n\ge n_0$, \[ c_{K,j}n^{-(d+1-j)} \le \mathbb E\left[U_j(K)-U_j(P_n)\right] \le C_{K,j}n^{-(d+1-j)}. \] The source does not explicitly define "of order"; the displayed interpretation requires confirmation. The exponent is retained exactly as supplied. A fully specified conjecture requires the ambient dimension, the definitions of $U_j$, the sampling convention, and any necessary regularity hypotheses to be checked. No smoothness or rolling-ball assumption has been added to the general question.

Short English statement

Sample $n$ independent uniform points from a spherically convex body contained in an open hemisphere, and take their spherical convex hull. The question asks whether the expected deficit in each $U_j$ has two-sided order $n^{-(d+1-j)}$. The dimension convention, the functionals $U_j$, the sampling measure, and any necessary additional hypotheses remain unresolved in the supplied record.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30003066$, title "Intrinsic-Volume Deficits of Random Spherical Polytopes," source problem code catalogue entry, collection "Oberwolfach Reports Open Problems." The proposed year is 2016; no individual proposer is supplied.
• Supplied report attribution: Asymptotic Geometric Analysis (2016), Oberwolfach Reports, pp. 507--565, DOI 10.4171/owr/2016/11. https://doi.org/10.4171/owr/2016/11. This report is the supplied workshop-problem provenance; no exact problem page is identified.
• Supplied literature source labelled "expected intrinsic volumes": https://doi.org/10.1007/s10998-008-8143-4. Authors, full bibliographic details, and applicable theorem statements are not included in the record.
• Supplied literature source labelled "boundary random polytopes": https://arxiv.org/abs/1103.4978. Authors, full bibliographic details, and applicable theorem statements are not included in the record.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem "partially solved" and states that the conjectured $n^{-(d+1-j)}$ expectation scale is established in important smooth or rolling-ball settings and for several intrinsic volumes, while precise asymptotic relations for the remaining indices are not known in full generality. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting that it is self-contained and preserves the target. These are supplied editorial claims; they do not resolve the missing definitions or constitute independent mathematical verification here.