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 Image may be NSFW.
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is bounded by Image may be NSFW.
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, for some universal constant Image may be NSFW.
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.
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 Image may be NSFW.
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of volume one, there exists a hyperplane Image may be NSFW.
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such that
Image may be NSFW.
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where Image may be NSFW.
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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).
Image may be NSFW.
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An artistic depiction of the concept “slice,” capturing a surreal and vibrant visualization. (ChatGPT)
Two recent posts of interest:
- Peter Sarnak will deliver the Mark Gordon memorial lecture series on Spectra of locally symmetric geometries at the Hebrew University of Jerusalem, on December 26 and 30 (2:30-3:30) and January 2 (10:30-11:30). For more, see Peter Sarnak is Coming to Town – Let’s Celebrate it with a Post on Möbius Randomness, Computational Complexity, and AI.
- The Case Against Google’s Claims of “Quantum Supremacy” explains why Google’s recent claim of “ten septillion years beyond classical” should be taken with decillion grains of salt Image may be NSFW.
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