{"paper":{"title":"On the distribution of distances in homogeneous compact metric spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GN"],"primary_cat":"math.MG","authors_text":"Jonathan Pakianathan, Mark Herman","submitted_at":"2014-07-21T19:33:35Z","abstract_excerpt":"We provide a simple proof that in any homogeneous, compact metric space of diameter $D$, if one finds the average distance $A$ achieved in $X$ with respect to some isometry invariant Borel probability measure, then $$\\frac{D}{2} \\leq A \\leq D.$$ This result applies equally to vertex-transitive graphs and to compact, connected, homogeneous Riemannian manifolds.\n  We then classify the cases where one of the extremes occurs. In particular any homogeneous compact metric space where $A=\\frac{D}{2}$ possesses a strict antipodal property which implies in particular that the distribution of distances "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.5607","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":""},"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"}