{"paper":{"title":"On Local Convexity Of Nonlinear Mappings Between Banach Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Anatolij Plichko, Anatoliy Prykarpatsky, Iryna Banakh, Taras Banakh","submitted_at":"2012-05-12T21:10:26Z","abstract_excerpt":"We find conditions for a smooth nonlinear map $f:U\\rightarrow V$ between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some $c$ and each positive $\\varepsilon<c$ the image $% f(B_\\varepsilon(x))$ of each $\\varepsilon$-ball $B_\\varepsilon(x)\\subset U$ is convex. We give a lower bound on $c$ via the second order Lipschitz constant $\\mathrm{Lip}_2(f)$, the Lipschitz-open constant $\\mathrm{Lip}_o(f)$ of $f$, and the 2-convexity number $\\mathrm{conv}_2(X)$ of the Banach space $% X$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.3412","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"}