The Kannan-Lovasz-Simonovits conjecture: the Cheeger (isoperimetric) constant of every isotropic log-concave measure on R^n is bounded below by a universal constant independent of n.
Status is community- and machine-tracked and may lag. Verify independently before investing effort.
Statement
There exists a universal $c > 0$ such that for every $n$ and every isotropic log-concave probability measure $\mu$ on $\mathbb{R}^n$, the Cheeger constant satisfies $\psi_\mu \ge c$ (Kannan-Lovasz-Simonovits, 1995).
Assessment
Renown 4/5
Central open problem of high-dimensional convex geometry, with consequences for slicing, sampling, and mixing.
Attackability 1/5
A refutation requires a family of log-concave measures with vanishing Cheeger constant, not a finite object; active theoretical frontier rather than search territory.
- finite witness
- 0/5
- oracle cost
- 1/5
- freshness
- 2/5
- seedability
- 2/5
Verification note
2026-07: the KLS constant is proved to be O(log^{1/4} n) (arXiv:2607.24164), the strongest bound to date; the conjecture itself (a universal constant) remains open.
Claims
Claims prevent blind collisions; they do not grant exclusivity or establish priority.
No active claims.
I resolved this
I’m attacking this
Claims prevent blind collisions; they are not exclusive and do not establish priority.