The Kannan–Lov\u00e1sz–Simonovits (KLS) conjecture asks whether every isotropic log-concave probability measure on Rn has a Cheeger constant bounded below by a universal positive constant. The best previous upper bound is \u03c8n\u2272log1\/4n, due to Letwin [Let26]. We prove that \u03c8n\u2264C\u22c54log\u2217(n+2) for a universal constant C, where log\u2217x is the least number of successive natural logarithms needed to bring x to at most one. We also prove that CP(\u03bc)\u2264C\u203216log\u2217(n+2) for every isotropic log-concave probability measure \u03bc on Rn, with a universal constant C\u2032>0.<\/p>\n","protected":false},"excerpt":{"rendered":"
The Kannan–Lov\u00e1sz–Simonovits (KLS) conjecture asks whether every isotropic log-concave probability measure on Rn has a Cheeger constant bounded below by a universal positive constant. The best previous upper bound is \u03c8n\u2272log1\/4n, due to Letwin [Let26]. We prove that \u03c8n\u2264C\u22c54log\u2217(n+2) for a universal constant C, where log\u2217x is the least number of successive natural logarithms needed […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":[{"type":"user_nicename","value":"Zhao Song","user_id":"37935"},{"type":"user_nicename","value":"Xinzhi 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