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

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.

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

1. Johan Aspegren says:

Those problems are trivial for any unit ball K , if one has that for the standard simplex S_n it holds that |S_n| < |K|. See https://math.stackexchange.com/questions/4297907/when-does-an-frame-of-rank-n-contain-n-orthonormal-vectors. Moreover, some asymptotic estimates are not ambitious enough. You wan' t sharp control for an unit ball K in any fixed dimension n, and describe the extremal cases.

2. Johan Aspegren says:

However, you need to do some scaling for S_n or K, to ensure that K and S_n have a comparable diameter.