{"data":{"id":"kls-conjecture-1995","statement_oneline":"The Kannan-Lovasz-Simonovits conjecture: the Cheeger (isoperimetric) constant of every isotropic log-concave measure on R^n is bounded below by a universal constant independent of n.","statement_full_latex":"There exists a universal $c > 0$ such that for every $n$ and every isotropic log-concave probability measure $\\mu$ on $\\mathbb{R}^n$, the Cheeger constant satisfies $\\psi_\\mu \\ge c$ (Kannan-Lovasz-Simonovits, 1995).","source":{"title":"Kannan-Lovasz-Simonovits (1995); survey literature","arxiv":null,"url":"https://en.wikipedia.org/wiki/Kannan%E2%80%93Lov%C3%A1sz%E2%80%93Simonovits_conjecture","year":1995,"area":"math.PR"},"renown":{"score":4,"rationale":"Central open problem of high-dimensional convex geometry, with consequences for slicing, sampling, and mixing.","scored_by":"founder"},"attackability":{"score":1,"rationale":"A refutation requires a family of log-concave measures with vanishing Cheeger constant, not a finite object; active theoretical frontier rather than search territory.","scored_by":"founder","subscores":{"finite_witness":0,"oracle_cost":1,"freshness":2,"seedability":2}},"status":"open","last_verified_open":"2026-07-28","added":"2026-07-28","verification_note":"2026-07: the KLS constant is proved to be O(log^{1/4} n) (arXiv:2607.24164), the strongest bound to date; the conjecture itself (a universal constant) remains open.","ranking_score":4,"verification_tier":null,"ai_systems":[],"posed_by":null,"years_open":null,"notability":null,"source_trackers":[],"resolution_type":null,"resolver":null,"resolution_date":null,"evidence_url":null,"evidence_note":null,"method_note":null,"effective_status":"open","claims":[],"status_events":[{"id":"founder-batch-20260728-kls-conjecture-1995","conjecture_id":"kls-conjecture-1995","status":"open","date":"2026-07-28T00:00:00.000Z","actor":"human-admin","rationale":"founder-curated batch, 2026-07-28","note":"Added open conjecture."}],"public_actions":[{"name":"claim","method":"POST","href":"/c/kls-conjecture-1995/claim","verification":"email magic link"},{"name":"report_resolution","method":"POST","href":"/c/kls-conjecture-1995/resolve","verification":"admin review"}],"resolutions":[]},"notice":"Status is community- and machine-tracked and may lag. Verify independently before investing effort."}