Selected research history · publication v4

Kannan–Lovász–Simonovits conjecture

Follow the ideas, results and limitations over time. This is a selected history, not a complete literature survey or a measure of how close the problem is to being solved.

Current state & remaining question →

Viewing the timeline from pinned publication v4.

4 selected milestones · Source coverage checked 2026-10-13 · Historical annotation and mathematical review remain separate.

1995 publication
Foundation
The convex-geometric conjecture
The cited manuscripts trace KLS to Kannan, Lovász and Simonovits’ isoperimetric work.
Scope & review: The reader uses the equivalent Poincaré formulation up to universal constants.
Statements & sources
- Presenting a proof of the Kannan–Lovász–Simonovits conjecture ↗Theorem 1.1 and proof/AI acknowledgement
- An O(4^log* n) Bound for the KLS Constant — first revision ↗Main theorem and Poincaré conversion; exact first revision
Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.
Oct 1, 2026 preprint
Proposal
A slowly dimension-dependent bound is proposed
Song and Zhang’s first revision states an inverse Cheeger estimate with iterated-logarithm dependence.
Scope & review: The corresponding Poincaré bound is squared and remains dimension dependent.
Statements & sources
- The first revision’s iterated-logarithm estimate →
- An O(4^log* n) Bound for the KLS Constant — first revision ↗Main theorem and Poincaré conversion; exact first revision
Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.
Oct 4, 2026 preprint
Proposal
Another manuscript presents a proof
Bizeul, Klartag and Lehec present a route through cumulants and suspension using a Song–Zhang criterion.
Scope & review: Different authors and arguments do not by themselves constitute independent validation.
Statements & sources
- A uniform Poincaré bound for isotropic log-concave measures →
- Presenting a proof of the Kannan–Lovász–Simonovits conjecture ↗Theorem 1.1 and proof/AI acknowledgement
Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.
Oct 4, 2026 preprint
Proposal
The second revision proposes a constant bound
Song and Zhang replace the earlier asymptotic target with a uniform bound.
Scope & review: This source revision supersedes the earlier passage but preserves its historical identity and statement.
Statements & sources
- A uniform Poincaré bound for isotropic log-concave measures →
- An O(1) Bound for the KLS Constant — second revision ↗Second revision main theorem and AI acknowledgement
Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

← Back to the Problem

The convex-geometric conjecture

A slowly dimension-dependent bound is proposed

Another manuscript presents a proof

The second revision proposes a constant bound