{"paper":{"title":"Reducing Isotropy and Volume to KLS: Faster Rounding and Volume Algorithms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math.FA"],"primary_cat":"cs.DS","authors_text":"Aditi Laddha, He Jia, Santosh S. Vempala, Yin Tat Lee","submitted_at":"2020-08-05T14:08:16Z","abstract_excerpt":"We show that the volume of a convex body in $\\mathbb{R}^{n}$ in the general membership oracle model can be computed to within relative error $\\varepsilon$ using $\\widetilde{O}(n^{3.5}\\psi^{2} + n^3/\\varepsilon^{2})$ oracle queries, where $\\psi$ is the KLS constant. With the current bound of $\\psi=\\widetilde{O}(1)$, this gives an $\\widetilde{O}(n^{3.5} + n^3/\\varepsilon^{2})$ algorithm, improving on the Lov\\'{a}sz-Vempala $\\widetilde{O}(n^{4}/\\varepsilon^{2})$ algorithm from 2003. The main new ingredient is an $\\widetilde{O}(n^{3}\\psi^{2})$ algorithm for isotropic transformation of a well-round"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2008.02146","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2008.02146/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}