{"id":1190158,"date":"2026-10-16T11:48:29","date_gmt":"2026-10-16T18:48:29","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=1190158"},"modified":"2026-10-16T11:48:30","modified_gmt":"2026-10-16T18:48:30","slug":"an-o1-bound-for-the-kls-constant","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/an-o1-bound-for-the-kls-constant\/","title":{"rendered":"An O(1) Bound for the KLS Constant"},"content":{"rendered":"\n\n\n<p class=\"wp-block-paragraph\">The Kannan\u2013Lov\u00b4asz\u2013Simonovits (KLS) conjecture asks whether every isotropic log-concave probability measure on <math><mrow><msup><mi>\u211d<\/mi><mi>n<\/mi><\/msup><\/mrow><\/math> has a Cheeger constant bounded below by a universal positive constant. We prove that <math><mrow><msub><mi>\u03c8<\/mi><mi>n<\/mi><\/msub><\/mrow><\/math> \u2264 <math><mi>C<\/mi><\/math> for a universal constant <math><mi>C<\/mi><\/math> > 0, resolving the KLS conjecture. We also prove that <math><mrow><msub><mi>C<\/mi><mi>P<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>\u03bc<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/math> \u2264 <math><mi>C<\/mi><\/math>\u2032 for every isotropic log-concave probability measure <math><mi>\u03bc<\/mi><\/math> on <math><mrow><msup><mi>\u211d<\/mi><mi>n<\/mi><\/msup><\/mrow><\/math>, with a universal constant <math><mi>C<\/mi><\/math>&#8216; > 0.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Kannan\u2013Lov\u00b4asz\u2013Simonovits (KLS) conjecture asks whether every isotropic log-concave probability measure on \u211dn has a Cheeger constant bounded below by a universal positive constant. We prove that \u03c8n \u2264 C for a universal constant C > 0, resolving the KLS conjecture. We also prove that CP(\u03bc) \u2264 C\u2032 for every isotropic log-concave probability measure \u03bc [&hellip;]<\/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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