Hilbert-Space Coarse Embeddings of Coarse Median Spaces

Definitions and mathematical statement

Use the following standard finite approximation convention for a coarse median space. A median algebra is a set $M$ with a symmetric ternary operation $m\colon M^3\to M$ satisfying, for all $a,b,c,d,e\in M$, \[ m(a,a,b)=a,\qquad m(m(a,b,c),d,e)=m(m(a,d,e),m(b,d,e),c). \] A metric space $(X,d)$ is coarse median if it admits an operation $\mu\colon X^3\to X$, a constant $K\geq 1$, and a function $h\colon{\mathbb{Z}}_{\geq0}\to[0,\infty)$ with the following properties. First, for all $x_1,x_2,x_3,y_1,y_2,y_3\in X$, \[ d\left(\mu(x_1,x_2,x_3),\mu(y_1,y_2,y_3)\right) \leq K\sum_{i=1}^3d(x_i,y_i)+h(0). \] Second, for every integer $p\geq1$ and every nonempty finite subset $A\subseteq X$ with $|A|\leq p$, there exist a finite median algebra $(M,m)$ and maps $\pi\colon A\to M$ and $\lambda\colon M\to X$ such that \[ d(a,\lambda\pi(a))\leq h(p)\quad(a\in A) \] and \[ d\left(\lambda m(u,v,w),\mu(\lambda u,\lambda v,\lambda w)\right)\leq h(p) \quad(u,v,w\in M). \] A coarse embedding into a Hilbert space is a map $f\colon X\to\mathcal H$ for which there exist nondecreasing functions $\rho_-,\rho_+\colon[0,\infty)\to[0,\infty)$ satisfying \[ \lim_{t\to\infty}\rho_-(t)=\infty,\qquad \rho_-(d(x,y))\leq\lVert f(x)-f(y)\rVert_{\mathcal H}\leq\rho_+(d(x,y)) \quad(x,y\in X). \] Question. Does every coarse median metric space $(X,d)$ admit a coarse embedding into some Hilbert space $\mathcal H$? The Hilbert space, map, and control functions may depend on $X$.

Unresolved source qualifications. The supplied record does not define coarse median space or specify finite rank, geodesicity, properness, bounded geometry, or separability assumptions. The question above uses the stated convention without imposing those additional assumptions; this interpretation requires mathematical review against the report. The original extraction also contains adjacent questions about symmetric groups and wall structures, attributed there to Arzhantseva, and single-peaked preferences, attributed there to Chatterji. These fragments are not incorporated into this question, and neither attribution is assigned to it.

Short English statement

Does every metric space with a coarse median operation admit a map into a Hilbert space whose distances have an upper bound depending only on the original distance and a lower bound tending to infinity with that distance? The source supplies the label "open"; its precise convention for coarse median spaces requires confirmation.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30006612$, title Hilbert-Space Coarse Embeddings of Coarse Median Spaces, source code catalogue entry, collection Oberwolfach Reports Open Problems. The repository registry locations are data/unsolvedmath/problems.json and catalog/problems.json; their contents were not independently inspected for this draft.
• Supplied report citation: Median Geometry and Applications (2026), Oberwolfach Reports, DOI $10.4171/owr/2026/8$, pp. 487--540. https://doi.org/10.4171/owr/2026/8. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No proposer is supplied for this question; the proposed year is 2026.
• Supplied literature source, labelled "OWR report": https://ems.press/content/serial-article-files/53603.
• Literature assessment checked on 2026-08-22: the supplied dated curation labels the question open, says the report directly poses the universal coarse-median-to-Hilbert coarse-embedding question and supplies no proof or counterexample, and reports that no resolution was found. This is supplied dated curation, not an independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its accompanying qualification says that only clear formula delimiters, notation, grammar, and typography were repaired and that mathematical content and scope were preserved. This records the supplied editorial assessment, not independent verification of this draft.