Wasserstein barycenters on metric graphs

Key Points

• The aim is to establish conditions under which a Wasserstein barycenter is absolutely continuous with respect to its Hausdorff measure.
• Examined Wasserstein barycenters on specific metric graphs.
• Analyzed the relationship between barycenters and Hausdorff measure.
• Identified key graph properties affecting absolute continuity.
• Showed that under certain conditions, barycenters are absolutely continuous away from vertices.
• Established criteria for the Hausdorff measure's behavior at barycenters.
• Provided examples illustrating the main findings.