{"paper":{"title":"Monotone and convex restrictions of continuous functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Zoltan Buczolich","submitted_at":"2016-07-12T09:21:45Z","abstract_excerpt":"Suppose that $f$ belongs to a suitably defined complete metric space $ {{\\cal C}}^{{\\alpha}}$ of H\\\"older $ {\\alpha}$-functions defined on $[0,1]$. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper Minkowski dimension) sets $A {\\subset} [0,1]$ such that $f|_{A}$ is monotone, or convex/concave. Some of our results are about generic functions in $ {{\\cal C}}^{{\\alpha}}$ like the following one: we prove that for the generic $f\\in C_{1}^{{\\alpha}}[0,1]$, $0\\leq {\\alpha}<2$ for any $A {\\subset} [0,1]$ such that $f|_{A}$ is convex, or concave we have ${\\mathr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.03279","kind":"arxiv","version":3},"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"}