{"paper":{"title":"Detecting fixed points of nonexpansive maps by illuminating the unit ball","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Bas Lemmens, Brian Lins, Roger Nussbaum","submitted_at":"2016-07-06T13:02:20Z","abstract_excerpt":"We give necessary and sufficient conditions for a nonexpansive map on a finite dimensional normed space to have a nonempty, bounded set of fixed points. Among other results we show that if $f : V \\rightarrow V$ is a nonexpansive map on a finite dimensional normed space $V$, then the fixed point set of $f$ is nonempty and bounded if and only if there exist $w_1, \\ldots , w_m$ in $V$ such that $\\{f(w_i) - w_i : i = 1, \\ldots, m \\}$ illuminates the unit ball. This yields a numerical procedure for detecting fixed points of nonexpansive maps on finite dimensional spaces. We also discuss application"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.01602","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"}