Logarithmic bounds for isoperimetry and slices of convex sets

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 a refined log-concave Lichnerowicz inequality.