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arXiv · 2412.15044

Affirmative Resolution of Bourgain's Slicing Problem using Guan's Bound

Boaz Klartag·Joseph Lehec
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Abstract

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 $K \subseteq \mathbb{R}^n$ of volume one, there exists a hyperplane $H \subseteq \mathbb{R}^n$ such that $$ Vol_{n-1}(K \cap H) > c, $$ where $c > 0$ is a universal constant. Our proof combines Milman's theory of $M$-ellipsoids, stochastic localization with a recent bound by Guan, and stability estimates for the Shannon-Stam inequality by Eldan and Mikulincer.

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Boaz Klartag, Joseph Lehec. 2024-12-19. Affirmative Resolution of Bourgain's Slicing Problem using Guan's Bound. https://arxiv.org/abs/2412.15044

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