{"paper":{"title":"On a Convex Operator for Finite Sets","license":"","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Branko \\'Curgus, Krzysztof Ko{\\l}odziejczyk","submitted_at":"2006-06-15T17:28:29Z","abstract_excerpt":"Let $S$ be a finite set with $n$ elements in a real linear space. Let $\\cJ_S$ be a set of $n$ intervals in $\\nR$. We introduce a convex operator $\\co(S,\\cJ_S)$ which generalizes the familiar concepts of the convex hull $\\conv S$ and the affine hull $\\aff S$ of $S$. We establish basic properties of this operator. It is proved that each homothet of $\\conv S$ that is contained in $\\aff S$ can be obtained using this operator. A variety of convex subsets of $\\aff S$ can also be obtained. For example, this operator assigns a regular dodecagon to the 4-element set consisting of the vertices and the o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0606371","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"}