{"paper":{"title":"On the structure of optimal free Dirichlet regions in mass transportation problems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Lucas D. O'Brien","submitted_at":"2026-06-29T18:56:52Z","abstract_excerpt":"For a compactly supported probability measure $\\mu$ on the $d$-dimensional space $\\mathbb{R}^d$, the average distance problem asks us to minimize the average distance functional over all compact, connected, $\\Sigma \\subseteq \\mathbb{R}^d$ satisfying the Hausdorff $1$-measure constraint $\\mathcal{H}^1(\\Sigma) \\leq \\ell$. This problem was first introduced in 2002 by Buttazzo, Oudet, and Stepanov to study optimal transport problems with free regions on which the transport cost vanishes, and has undergone a considerable amount of research since. Most recently, Kobayashi, Kim, and the author studie"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.30826","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.30826/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}