Smeariness, Stability, and Strong Consistency on Manifolds

Definitions and mathematical statement

The supplied record groups three questions concerning smeariness, stability, stickiness, and strong consistency. Several essential definitions and hypotheses are absent, so the following preserves the questions while explicitly identifying the unresolved qualifications.

1. For a non-positively curved manifold, determine whether only non-positive $k$-smeariness can occur. The record does not define $k$-smeariness, specify the range of $k$, identify the statistical functional or probability measures involved, or specify the notion of curvature. Moreover, the wording does not distinguish a universal assertion excluding positive $k$-smeariness on every such manifold from an existence question about a manifold exhibiting only non-positive $k$-smeariness. These alternatives cannot be resolved from the supplied record.
2. Determine the relationship between manifold stability and stickiness on a non-Hausdorff Lie-group quotient. The acting Lie group, the space acted upon, the action, the quotient topology, the statistical functional, and the definitions of manifold stability and stickiness are not supplied. Consequently, no precise implication, equivalence, or existence assertion is specified by this question.
3. Determine whether the inclusion attributed in the record to Ziezold's strong consistency result can be strict: \[ \bigcup_{n=1}^{\infty}\bigcap_{k=n}^{\infty}\widehat E_\rho^{(k)}\subset E_\rho. \] Here the symbol $\subset$ is read as inclusion without asserting strictness, as required by the question. For the purpose of interpreting this displayed set expression only, let $X$ be a common ambient set containing $E_\rho$ and every $\widehat E_\rho^{(k)}$, with $n,k$ positive integers. Its left-hand side is the set-theoretic lower limit \[ \liminf_{k\to\infty}\widehat E_\rho^{(k)} =\left\{x\in X:\exists n\geq 1\ \forall k\geq n,\ x\in\widehat E_\rho^{(k)}\right\}. \] Thus the requested strictness means \[ \liminf_{k\to\infty}\widehat E_\rho^{(k)}\subseteq E_\rho, \qquad E_\rho\setminus\liminf_{k\to\infty}\widehat E_\rho^{(k)}\neq\varnothing. \] The record does not define $\rho$, $E_\rho$, or $\widehat E_\rho^{(k)}$, or supply the hypotheses of the attributed result. If these sets depend on random observations, the sampling assumptions and the probability qualification for the inclusion and its possible strictness are also missing. The displayed expression is preserved literally; it is not replaced by a topological limit or by an expression involving closures.

A fully quantified formulation requires the missing source definitions and hypotheses. No resolution of these questions is asserted here.

Short English statement

The record asks whether non-positive curvature restricts smeariness, how stability and stickiness relate on a non-Hausdorff Lie-group quotient, and whether a set inclusion attributed to Ziezold can be strict. Essential definitions and hypotheses are missing, so these are qualified statement-only drafts requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002682$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. Project registry references: data/unsolvedmath/problems.json and catalog/problems.json. This draft uses the supplied record; the registry files were not independently inspected.
• Supplied report citation: Mini-Workshop: Asymptotic Statistics on Stratified Spaces (2015), Oberwolfach Reports, pp. 2481--2527, DOI 10.4171/owr/2014/44. https://doi.org/10.4171/owr/2014/44. The literature-source label is Oberwolfach Report 44/2014. The record gives proposed year 2014 but includes 2015 in the report title; this discrepancy is unresolved. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Sticky Central Limit Theorems on Open Books. https://doi.org/10.1214/12-AAP899.
• Supplied literature reference: Central Limit Theorems for Frechet Means. https://doi.org/10.48550/arXiv.2311.09455.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports rigorous laws of large numbers and central limit theorems for sticky means in open books, cones, treespaces, and broader stratified settings, while describing the listed curvature, quotient-stability, and strong-consistency questions as unsettled in full generality. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.
• The strong-consistency inclusion is attributed to Ziezold by the supplied statement. No specific bibliographic citation or theorem locator for that result, and no individual proposer for the combined problem, is supplied. The registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication; this is a supplied editorial qualification, not independent verification by this draft.