Bourgain's slicing problem and KLS isoperimetry up to polylog

TL;DR

This paper demonstrates that Bourgain's slicing problem and the KLS isoperimetric conjecture are valid up to polylogarithmic factors in high dimensions, advancing understanding in geometric functional analysis.

Contribution

The authors prove that both conjectures hold true up to polylogarithmic factors, providing significant progress in high-dimensional convex geometry.

Findings

Bourgain's slicing problem is confirmed up to polylogarithmic factors.

KLS isoperimetric conjecture is validated up to polylogarithmic factors.

The results improve bounds in high-dimensional convex geometry.

Abstract

We prove that Bourgain's hyperplane conjecture and the Kannan-Lov\'asz-Simonovits (KLS) isoperimetric conjecture hold true up to a factor that is polylogarithmic in the dimension.