{"paper":{"title":"The geometry of two-valued subsets of $L_{p}$-spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Anthony Weston","submitted_at":"2014-12-29T21:01:46Z","abstract_excerpt":"Let $\\mathcal{M}(\\Omega, \\mu)$ denote the algebra of all scalar-valued measurable functions on a measure space $(\\Omega, \\mu)$. Let $B \\subset \\mathcal{M}(\\Omega, \\mu)$ be a set of finitely supported measurable functions such that the essential range of each $f \\in B$ is a subset of $\\{ 0,1 \\}$. The main result of this paper shows that for any $p \\in (0, \\infty)$, $B$ has strict $p$-negative type when viewed as a metric subspace of $L_{p}(\\Omega, \\mu)$ if and only if $B$ is an affinely independent subset of $\\mathcal{M}(\\Omega, \\mu)$ (when $\\mathcal{M}(\\Omega, \\mu)$ is considered as a real vec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.8481","kind":"arxiv","version":4},"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"}