{"paper":{"title":"Floating body, illumination body, and polytopal approximation","license":"","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Carsten Sch\\\"utt","submitted_at":"1996-09-05T00:00:00Z","abstract_excerpt":"Let $K$ be a convex body in $\\Bbb R^{d}$ and $K_{t}$ its floating bodies. There is a polytope with at most $n$ vertices that satisfies $$ K_{t} \\subset P_{n} \\subset K $$ where $$ n \\leq e^{16d} \\frac{vol_{d}(K \\setminus K_{t})}{t\\ vol_{d}(B_{2}^{d})} $$ Let $K^{t}$ be the illumination bodies of $K$ and $Q_{n}$ a polytope that contains $K$ and has at most $n$ $d-1$-dimensional faces. Then $$ vol_{d}(K^{t} \\setminus K) \\leq cd^{4} vol_{d}(Q_{n} \\setminus K) $$ where $$ n \\leq \\frac{c}{dt} \\ vol_{d}(K^{t} \\setminus K) $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9609206","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"}