{"paper":{"title":"Polygonal equalities and virtual degeneracy in $L_{p}$-spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Anthony Weston, Casey Lynn Kelleher, Daniel Miller, Trenton Osborn","submitted_at":"2012-03-26T22:53:47Z","abstract_excerpt":"Suppose $0 < p \\leq 2$ and that $(\\Omega, \\mu)$ is a measure space for which $L_{p}(\\Omega, \\mu)$ is at least two-dimensional. The central results of this paper provide a complete description of the subsets of $L_{p}(\\Omega, \\mu)$ that have strict $p$-negative type. In order to do this we study non-trivial $p$-polygonal equalities in $L_{p}(\\Omega, \\mu)$. These are equalities that can, after appropriate rearrangement and simplification, be expressed in the form \\begin{eqnarray*} \\sum\\limits_{j, i = 1}^{n} \\alpha_{j} \\alpha_{i} {\\| z_{j} - z_{i} \\|}_{p}^{p} & = & 0 \\end{eqnarray*} where $\\{ z_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.5837","kind":"arxiv","version":6},"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"}