{"paper":{"title":"Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CG","authors_text":"Eleni Tzanaki, Menelaos I. Karavelas","submitted_at":"2009-11-26T20:06:03Z","abstract_excerpt":"Given a set $\\Sigma$ of spheres in $\\mathbb{E}^d$, with $d\\ge{}3$ and $d$ odd, having a fixed number of $m$ distinct radii $\\rho_1,\\rho_2,...,\\rho_m$, we show that the worst-case combinatorial complexity of the convex hull $CH_d(\\Sigma)$ of $\\Sigma$ is $\\Theta(\\sum_{1\\le{}i\\ne{}j\\le{}m}n_in_j^{\\lfloor\\frac{d}{2}\\rfloor})$, where $n_i$ is the number of spheres in $\\Sigma$ with radius $\\rho_i$.\n  To prove the lower bound, we construct a set of $\\Theta(n_1+n_2)$ spheres in $\\mathbb{E}^d$, with $d\\ge{}3$ odd, where $n_i$ spheres have radius $\\rho_i$, $i=1,2$, and $\\rho_2\\ne\\rho_1$, such that their"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.5086","kind":"arxiv","version":5},"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"}