An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture
Yuansi Chen
Abstract
Abstract We prove an almost constant lower bound of the isoperimetric coefficient in the KLS conjecture. The lower bound has the dimension dependency $$d^{-o_d(1)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>d</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:msub><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup></mml:math> . When the dimension is large enough, our lower bound is tighter than the previous best bound which has the dimension dependency $$d^{-1/4}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>d</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:math> . Improving the current best lower bound of the isoperimetric coefficient in the KLS conjecture has many implications, including improvements of the current best bounds in Bourgain’s slicing conjecture and in the thin-shell conjecture, better concentration inequalities for Lipschitz functions of log-concave measures and better mixing time bounds for MCMC sampling algorithms on log-concave measures.