{"paper":{"title":"Hyperbolic $p$-barycenters, circumcenters, and Moebius maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Kingshook Biswas","submitted_at":"2017-11-06T17:37:22Z","abstract_excerpt":"Given a Moebius homeomorphism $f : \\partial X \\to \\partial Y$ between boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, and a family of probability measures $\\{ \\mu_x \\}_{x \\in X}$ on $\\partial X$, we describe a continuous family of extensions $\\{\\hat{f}_p : X \\to Y \\}_{1 \\leq p \\leq \\infty}$ of $f$, called the hyperbolic $p$-barycenter maps of $f$. If all the measures $\\mu_x$ have full support then for $p = \\infty$ the map $\\hat{f}_{\\infty}$ coincides with the circumcenter map $\\hat{f}$ defined previously in \\cite{biswas5}. We use this to show that if $X, Y$ are complete, simp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.02559","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"}