{"paper":{"title":"Optimal Approximate Polytope Membership","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"David M. Mount, Guilherme D. da Fonseca, Sunil Arya","submitted_at":"2016-12-06T08:07:36Z","abstract_excerpt":"In the polytope membership problem, a convex polytope $K$ in $R^d$ is given, and the objective is to preprocess $K$ into a data structure so that, given a query point $q \\in R^d$, it is possible to determine efficiently whether $q \\in K$. We consider this problem in an approximate setting and assume that $d$ is a constant. Given an approximation parameter $\\varepsilon > 0$, the query can be answered either way if the distance from $q$ to $K$'s boundary is at most $\\varepsilon$ times $K$'s diameter. Previous solutions to the problem were on the form of a space-time trade-off, where logarithmic "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.01696","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"}