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The slicing conjecture for convex bodies

About 9 years old · traced to

For a convex body K⊂Rn\mathcal{K}\subset\mathbb{R}^{n}, let vol⁡n−1(K∩H)\operatorname{vol}_{n-1}(\mathcal{K}\cap H) denote the (n−1)(n-1)-dimensional volume of its section by a hyperplane HH. Define the slicing constant LnL_n by

1Ln:=inf⁡K⊂Rnsup⁡Hvol⁡n−1(K∩H),\frac{1}{L_n}:=\inf_{\mathcal{K}\subset\mathbb{R}^{n}}\sup_H\operatorname{vol}_{n-1}(\mathcal{K}\cap H),

where the infimum is over convex bodies of volume one and the supremum is over hyperplanes.

Slicing conjecture. The slicing constant is universally bounded:

Ln=O(1).L_n=O(1).

Following Guan's analysis, the source states that this conjecture was resolved by Klartag and Lehec and by Bizeul.

References

Primary source

Yunbum Kook and Santosh S. Vempala, “The Localization Method for High-Dimensional Inequalities”, arXiv:2512.10848 (2026).

Additional references

3 papers in this index state this conjecture (2017–2025). The statement above is taken from the most recent of them; the others are arXiv:2406.07406, arXiv:1707.03809.

convex-bodiesconvex-geometryslicing-problem
Posted

Progress summary

Refreshed
Claimed solved

A 2024–2025 proof announcement says the conjecture is true in every dimension, but this report records the resolution as unverified.

Bourgain posed the slicing problem in 1986: every volume-one convex body should have a hyperplane section whose size is bounded below independently of dimension. Recent sources attribute an affirmative resolution to Joseph Lehec and Bo’az Klartag, building on Qingyang Guan’s work, with an alternative proof by Pierre Bizeul.

Known results

  • Bourgain obtained LK=O(n1/4log⁡n)L_K=O(n^{1/4}\log n).
  • Klartag improved this to LK=O(n1/4)L_K=O(n^{1/4}).
  • Earlier work verified the conjecture for several classes, including ℓp\ell_p balls.
  • Yuansi Chen’s 2021 result gave an approximately constant slicing bound through implications from the KLS problem, without proving the full KLS conjecture.

Affirmative resolution claim, December 2024–January 2025

On December 20, 2024, Gil Kalai reported that Guan, Lehec, and Klartag had completed the argument. Bizeul’s preprint, version 1 dated January 12, 2025 and manuscript dated January 14, presents an alternative proof and states sup⁡n≥1Ln<+∞\sup_{n\ge 1}L_n<+\infty. A later catalogued source also records the conjecture as resolved, but the retrieved material contains no independent verification or published referee assessment.

Current status (as of September 2026): The conjecture is claimed solved by Klartag–Lehec, with Bizeul’s alternative proof, but the retrieved evidence does not independently verify the resolution.

Sources

Solutions 0

No solutions have been posted yet.