Bo’az Klartag and Joseph Lehec: The Slice Conjecture Up to Polylogarithmic Factor!

Combinatorics and more 2022-10-15

BoazJoseph

Bo’az Klartag (right) and Joseph Lehec (left)

In December 2020, we reported on Yuansi Chen breakthrough result on Bourgain’s alicing problem and the Kannan Lovasz Simonovits conjecture. It is a pleasure to report on a further fantastic progress on these problems.

Bourgain’s slicing problem (1984): Is there c > 0 such that for any dimension n and any centrally symmetric convex body K ⊆ $\mathbb R^n$ of volume one, there exists a hyperplane H such that the (n − 1)-dimensional volume of K ∩ H is at least c?

Some time ago we reported on Yuansi Chen’s startling result that c can be taken as $n^{-o(1)}$ . More precisely, Chen proved:

Theorem (Chen, 2021): For any dimension n and any centrally symmetric convex body K ⊆ $\mathbb R^n$ of volume one, there exists a hyperplane H such that the (n − 1)-dimensional volume of K ∩ H is at least $\frac {1}{L_n}$ where

$L_n=C \exp ( \sqrt {\log n} \sqrt {3\log \log n} ).$

A major improvement was recently achieved by Bo’az Klartag and Joseph Lehec

Theorem (Klartag and Lehec, 2022): For any dimension n and any centrally symmetric convex body K ⊆ $\mathbb R^n$ of volume one, there exists a hyperplane H such that the (n − 1)-dimensional volume of K ∩ H is at least $\frac {1}{L_n}$ where

$L_n = C (\log n)^4.$

Klartag and Lehec’s argument (as Chen’s earlier argument) relies on Ronen Eldan’s stochastic localization, with a new ingredients being the functional-analytic approach from a paper by Klartag and Eli Putterman

This is fantastic progress. Congratulations Bo’az and Joseph!

Update: I was happy to learn that Arun Jambulapati, Yin Tat Lee, and Santosh S. Vempala further improved in this paper the exponent from 4 to 2.2226.