{"paper":{"title":"A \"hidden\" characterization of approximatively polyhedral convex sets in Banach spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.FA","authors_text":"Ivan Hetman, Taras Banakh","submitted_at":"2011-11-29T07:01:34Z","abstract_excerpt":"For a Banach space $X$ by $Conv_H(X)$ we denote the space of non-empty closed convex subsets of $X$, endowed with the Hausdorff metric. We prove that for any closed convex set $C\\subset X$ and its metric component $H_C=\\{A\\in Conv_H(X):d_H(A,C)<\\infty\\}$ in $Conv_H(X)$, the following conditions are equivalent: (1) $C$ is approximatively polyhedral, which means that for every $\\epsilon>0$ there is a polyhedral convex subset $P\\subset X$ on Hausdorff distance $d_H(P,C)<\\epsilon$ from $C$; (2) $C$ lies on finite Hausdorff distance $d_H(C,P)$ from some polyhedral convex set $P\\subset X$; (3) the m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.6708","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"}