Affirmative Resolution of Bourgain’s Slicing Problem Using Guan’s Bound

We provide the final step in the resolution of Bourgain’s slicing problem in the affirmative. Thus we establish the following theorem: for any convex body K \subseteq \mathbb{R}^{n}$K \subseteq \mathbb{R}^{n}$ of volume one, there exists a hyperplane H \subseteq \mathbb{R}^{n}$H \subseteq \mathbb{R}^{n}$ such that

Vol_{n-1}(K \cap H) > c, $$Vol_{n-1}(K \cap H) > c,$$

where c > 0$c > 0$ is a universal constant. Our proof combines Milman’s theory of M$M$-ellipsoids, stochastic localization with a recent bound by Guan, and stability estimates for the Shannon-Stam inequality by Eldan and Mikulincer.