  We prove that the Bourgain slicing conjecture and the
Kannan-Lov\'asz-Simonovits (KLS) isoperimetric conjecture in $\mathbb{R}^n$
hold true up to a factor of $\sqrt{\log n}$. A new ingredient used in the proof
is an improved log-concave Lichnerowicz inequality.

Functional Analysis;Metric Geometry;Probability