Qingyang Guan, Joseph Lehec and Bo’az Klartag Solved The Slice Conjecture!

Combinatorics and more 2024-12-20

Good news: the slice conjecture is now completely settled with combined effort of two separate papers.

Qingyang Guan, A note on Bourgain’s slicing problem

Abstract: This note is to study Bourgain’s slicing problem following the routes investigated in the last decade. We show that the slicing constant $L_n$ is bounded by $C \log \log n, n\ge 3$ , for some universal constant $C$ .

Boaz Klartag and Joseph Lehec, Affirmative Resolution of Bourgain’s Slicing Problem using Guan’s Bound

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 $K \subset \mathbb R^n$ of volume one, there exists a hyperplane $H \subset \mathbb R^n$ such that

${\rm Vol}_{n-1} (L \cap H) > c$

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.

For earlier posts on the slice conjecture see: To Cheer You Up in Difficult Times 15: Yuansi Chen Achieved a Major Breakthrough on Bourgain’s Slicing Problem and the Kannan, Lovász and Simonovits Conjecture (2020); Bo’az Klartag and Joseph Lehec: The Slice Conjecture Up to Polylogarithmic Factor! (2022).

An artistic depiction of the concept “slice,” capturing a surreal and vibrant visualization. (ChatGPT)

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