{"paper":{"title":"Linear Functions to the Extended Reals","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["cs.GT","stat.TH"],"primary_cat":"math.ST","authors_text":"Bo Waggoner","submitted_at":"2021-02-18T18:53:53Z","abstract_excerpt":"This paper investigates functions from $\\mathbb{R}^d$ to $\\mathbb{R} \\cup \\{\\pm \\infty\\}$ that satisfy axioms of linearity wherever allowed by extended-value arithmetic. They have a nontrivial structure defined inductively on $d$, and unlike finite linear functions, they require $\\Omega(d^2)$ parameters to uniquely identify. In particular they can capture vertical tangent planes to epigraphs: a function (never $-\\infty$) is convex if and only if it has an extended-valued subgradient at every point in its effective domain, if and only if it is the supremum of a family of \"affine extended\" funct"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.09552","kind":"arxiv","version":2},"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/2102.09552/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"}