Path connectivity of the infinity Z-Gromov–Wasserstein space
Let $(Z, d_Z)$ be a nonempty, complete, separable, path-connected metric space. A $Z$-valued $\infty$-measure network is a triple $\mathcal{X} = (X, \omega_X, \mu_X)$, where $X$ is a Polish space, $\mu_X$ is a Borel probability measure on $X$, and $\omega_X : X \times X \to Z$ is measurable and essentially bounded: for some $z_0 \in Z$, $$\operatorname*{ess\,sup}_{\mu_X \otimes \mu_X} d_Z(\omega_X, z_0) < \infty.$$ No symmetry or metric axioms are imposed on the kernel $\omega_X$. For two such networks let $$\operatorname{GW}^Z_\infty(\mathcal{X}, \mathcal{Y}) = \frac12 \inf_{\pi \in \Pi(\mu_X, \mu_Y)} \operatorname*{ess\,sup}_{\pi \otimes \pi} d_Z\bigl(\omega_X(x, x'), \omega_Y(y, y')\bigr),$$ where $\Pi(\mu_X, \mu_Y)$ is the set of couplings of $\mu_X$ and $\mu_Y$, and let $\mathcal{M}_\infty(Z)$ be the metric space obtained by identifying networks at distance zero. Bauer, Mémoli, Needham and Nishino proved that $\operatorname{GW}^Z_\infty$ is a metric on networks up to weak isomorphism, that for every finite exponent $p$ the analogous space of $p$-networks is contractible whatever the topology of $Z$, and that $\mathcal{M}_\infty(Z)$ is geodesic when $Z$ is. For $Z = \{0, 1\} \subset \mathbb{R}$ they showed that $\mathcal{M}_\infty(Z)$ is not path connected.
Problem. Must $\mathcal{M}_\infty(Z)$ be path connected for every such $Z$, that is, can any two of its points be joined by a path $[0, 1] \to \mathcal{M}_\infty(Z)$ continuous for $\operatorname{GW}^Z_\infty$? The space $Z$ need not be geodesic, and each network along the path must itself be essentially bounded.
Source: Bauer, Mémoli, Needham and Nishino (2025), Question 41.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- GN, PR
- Keywords
- AIM Open Applied Problems, Gromov–Wasserstein distance, measure networks, path connectedness
References
@article{bauer2025,
author = {Bauer, Martin and M{\'e}moli, Facundo and Needham, Tom and Nishino, Mao},
title = {The {$Z$}-{G}romov--{W}asserstein distance},
journal = {J. Mach. Learn. Res.},
volume = {26},
number = {291},
year = {2025},
pages = {1--57},
url = {https://www.jmlr.org/papers/v26/24-2189.html},
eprint = {2408.08233},
archivePrefix = {arXiv},
}
@misc{aim-z-gw-infinity-path-connectivity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Path connectivity of the infinity Z-Gromov–Wasserstein space}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/research/resolved/661-z-gw-infinity-path-connectivity.md},
note = {Entry 661 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
No: for Z = R with the metric min{1, |s - t|}, M_infinity(Z) is disconnected, hence not path connected..
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, 514-infinity-z-gw/PROOF.md. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00739, "Path connectivity of the infinity Z-Gromov–Wasserstein space", clopen.solutions.