Coarse Median Spaces with $\delta$-Median Metrics

Definitions and mathematical statement

The supplied question is: Does every coarse median space admit a $\delta$-median metric? Its precise interpretation is unresolved because the excerpt contains inconsistent interval notation and does not specify how the new metric must relate to the original metric.

For a standard working convention, a coarse median structure on a metric space $(X,d)$ is a map $\mu\colon X^3\to X$ with constants $K,C\geq 0$ and a function $H\colon{\mathbb{N}}_{\geq 1}\to[0,\infty)$ satisfying the following conditions. First, \[ d\left(\mu(x_1,x_2,x_3),\mu(y_1,y_2,y_3)\right) \leq K\sum_{i=1}^3 d(x_i,y_i)+C. \] Second, for every nonempty finite $A\subseteq X$ with $|A|\leq p$, there are a finite median algebra $(M,m_M)$ and maps $\pi\colon A\to M$ and $\lambda\colon M\to X$ such that \[ d(a,\lambda\pi(a))\leq H(p)\qquad(a\in A) \] and \[ d\left(\lambda m_M(u,v,w),\mu(\lambda u,\lambda v,\lambda w)\right) \leq H(p)\qquad(u,v,w\in M). \] Here a finite median algebra may be represented as a nonempty subset of $\{0,1\}^n$, for some finite $n$, closed under coordinatewise majority, with $m_M$ the restricted majority operation. This is an explicit working convention; the supplied excerpt does not itself define coarse median spaces or specify a convention.

For a metric $d'$ on $X$ and $\delta\geq 0$, define the usual coarse interval by \[ I^{d'}_\delta(x,y) =\{w\in X:d'(x,w)+d'(w,y)\leq d'(x,y)+\delta\}. \] The excerpt does not supply a definition of $I_\delta$, so this formula is also an explicitly stated interpretive convention.

If the intended operation is triple intersection, the approximate-median condition can be expressed as follows: there exist finite constants $\delta,R\geq 0$ such that, for every $x,y,z\in X$, there exists $m\in X$ for which \[ \varnothing\neq S^{d'}_\delta(x,y,z) :=I^{d'}_\delta(x,y)\cap I^{d'}_\delta(y,z)\cap I^{d'}_\delta(z,x) \subseteq \{w\in X:d'(w,m)\leq R\}. \] Both constants are independent of $x,y,z$. Nonemptiness alone does not state this uniform boundedness requirement.

With these conventions, the intended question has the conditional schema \[ \begin{gathered} \text{For every coarse median space }(X,d,\mu),\text{ does there exist a metric }d'\text{ on }X\\ \text{satisfying the intended compatibility with }d\text{ and, if required, }\mu,\\ \text{together with }\delta,R\geq 0\text{ satisfying the displayed condition?} \end{gathered} \] The compatibility requirement is missing from the supplied record. In particular, the record does not say whether $d'$ must be bilipschitz equivalent, quasi-isometric, or otherwise related to $d$, or whether its approximate medians must agree coarsely with $\mu$. These possibilities are not interchangeable.

There is also a direct textual conflict: the displayed source definition uses \[ I_\delta(x,y)\cup I_\delta(y,z)\cup I_\delta(z,x)\sim_d\{m\}, \] where $\sim_d$ is described as meaning nonempty and at uniformly bounded distance from the singleton, whereas the subsequent prose refers to the triple intersection of the coarse intervals. The intersection formulation above is therefore a candidate interpretation, not an established correction of the source. A unique precise formulation requires resolution of this conflict and of the missing metric compatibility requirement.

Short English statement

The question asks whether every coarse median space can be equipped with a compatible metric whose three pairwise coarse intervals have a nonempty, uniformly bounded common intersection. This is only a candidate reading: the supplied display uses a union, the prose uses an intersection, and the required relationship between the original and replacement metrics is unspecified. The registry labels the question open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID 30005179, exact source problem code catalogue entry, title Coarse Median Spaces with $\delta$-Median Metrics. The record assigns proposed year 2022, names no proposer, labels the problem open, and gives statement status corrected_verified. Its statement-verification field reports reconstruction and checking against the official report; these are supplied registry assertions, not independent verification here.
• Supplied report citation: Topologie (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/34, pp. 1927--1986. https://doi.org/10.4171/owr/2022/34. The background and literature-source list also identify this as Oberwolfach Report 34/2022. Both supplied year descriptions are retained without reconciliation.
• Supplied literature source: Niblo--Wright--Zhang, Coarse median algebras. https://arxiv.org/abs/1802.02499. The excerpt attributes to Niblo, Wright and Zhang, through its reference [6], the assertion that $\delta$-median spaces are coarse median. That attribution is retained as a source claim.
• The excerpt mentions a paper with Cornelia Drutu as reference [1] and a result of Petyt as reference [7]. Their full bibliographic entries are not supplied. It defines a $\delta$-median group as one acting geometrically on a $\delta$-median space and attributes to Petyt an inclusion of mapping class groups of closed surfaces. It also lists cocompact lattices in products of rank-one Lie groups and non-cocompact lattices in $\mathrm{SO}(n,1)$, followed by a corrupted clause about lattices in $\mathrm{SL}(2,\cdot)$. These background claims are not independently verified, and the last group cannot be recovered reliably from the supplied text.
• Supplied literature assessment checked at 2026-08-22: the excerpt asserts the implication from $\delta$-median to coarse median, while the converse metric-existence question remains unclear; related metric-reconstruction results are said not to settle the stronger assertion. This is supplied dated curation, not an independently verified literature assessment. The record explicitly calls for independent verification of time-sensitive status and cited claims.