{"paper":{"title":"Structure of average distance minimizers in general dimensions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.OC","authors_text":"Forest Kobayashi, Lucas O'Brien, Young-Heon Kim","submitted_at":"2025-03-29T23:48:31Z","abstract_excerpt":"For a fixed, compactly supported probability measure $\\mu$ on the $d$-dimensional space $\\mathbb{R}^d$, we consider the problem of minimizing the $p^{\\mathrm{th}}$-power average distance functional over all compact, connected $\\Sigma \\subseteq \\mathbb{R}^d$ with Hausdorff 1-measure $\\mathcal{H}^1(\\Sigma) \\leq l$. This problem, known as the average distance problem, was first studied by Buttazzo, Oudet, and Stepanov in 2002, and has undergone a considerable amount of research since. We will provide a novel approach to studying this problem by analyzing it using the so-called \\textit{barycentre "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.23256","kind":"arxiv","version":3},"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/2503.23256/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"}