{"paper":{"title":"Approximations of convex bodies by measure-generated sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Boaz A. Slomka, Han Huang","submitted_at":"2017-06-21T20:21:16Z","abstract_excerpt":"Given a Borel measure $\\mu$ on ${\\mathbb R}^{n}$, we define a convex set by \\[ M({\\mu})=\\bigcup_{\\substack{0\\le f\\le1,\\\\ \\int_{{\\mathbb R}^{n}}f\\,{\\rm d}{\\mu}=1 } }\\left\\{ \\int_{{\\mathbb R}^{n}}yf\\left(y\\right)\\,{\\rm d}{\\mu}\\left(y\\right)\\right\\} , \\] where the union is taken over all $\\mu$-measurable functions $f:{\\mathbb R}^{n}\\to\\left[0,1\\right]$ with $\\int_{{\\mathbb R}^{n}}f\\,{\\rm d}{\\mu}=1$. We study the properties of these measure-generated sets, and use them to investigate natural variations of problems of approximation of general convex bodies by polytopes with as few vertices as possi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.07112","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"}