{"paper":{"title":"On averaged self-distances in finite dimensional Banach spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Gyula Lakos","submitted_at":"2024-11-21T13:56:33Z","abstract_excerpt":"Assume that $\\mathfrak A$ is a real Banach space of finite dimension $n\\geq2$. Consider any Borel probability measure $\\nu$ supported on the unit ball $K$ of $\\mathfrak A$. We show that \\[\\Delta(\\nu)=\\int_{x \\in K}\\int_{ y\\in K}|x-y|_{\\mathfrak A} \\,\\,\\,\\nu(x)\\,\\nu(y)\\leq 2(1-2^{-n}f(n)),\\] where $f:\\mathbb N\\setminus \\{0,1\\}\\rightarrow (0,1]$ is a concrete universal function such that $f(n)\\sim \\frac{2}{\\mathrm e n^2\\log n}$. It is hoped that in the estimate`$f(n)$' can be replaced by `$1$'."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.14129","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/2411.14129/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"}