{"paper":{"title":"One-radius results for supermedian functions on $\\Bbb R^d$, $d\\le 2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Nikolai Nikolov, Wolfhard Hansen","submitted_at":"2009-08-10T17:12:41Z","abstract_excerpt":"A classical result states that every lower bounded superharmonic function on $\\Bbb R^2$ is constant. In this paper the following (stronger) one-circle version is proven. If $f\\colon \\Bbb R^2\\to (-\\infty,\\infty]$ is lower semicontinuous, $\\liminf_{|x|\\to\\infty} f(x)/\\ln|x|\\ge 0$, and, for every $x\\in\\Bbb R^2$, $1/(2\\pi) \\int_0^{2\\pi} f(x+r(x)e^{it}) dt\\le f(x)$, where $r\\colon \\Bbb R^2\\to (0,\\infty)$ is continuous, $\\sup_{x\\in\\Bbb R^2} (r(x)-|x|)<\\infty$, and $\\inf_{x\\in\\Bbb R^2} (r(x)-|x|)=-\\infty$, then $f$ is constant.\n  Moreover, it is shown that, with respect to the assumption $r\\le c|\\cdo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.1251","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"}