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

TL;DR

The paper resolves Bourgain's slicing problem by proving a universal lower bound on the maximal hyperplane slice of any volume-one convex body, i.e. $Vol_{n-1}(K\cap H) > c$ for some universal $c>0$. It combines Milman's $M$-ellipsoid technique with heat-flow based stochastic localization and Guan's bound on the covariance process, together with stability results for the Shannon–Stam inequality via Eldan–Mikulincer and de Bruijn’s identity, to transfer isotropic constants across dimensions. A key dimension-reduction step yields a lower-dimensional isotropic measure $\nu$ with $L_{\nu} \gtrsim L_n$ and a nontrivial covariance mass along the flow, enabling entropy bounds to control isotropic constants. The final synthesis shows $\sup_n L_n < \infty$, thereby affirming Bourgain's hyperplane conjecture and strengthening the link between convex geometry and information-theoretic methods.

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.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3: Guan
• Proposition 2.4
• Lemma 2.5
• proof