The Kannan-Lovász-Simonovits conjecture, also called the KLS conjecture, asserts a dimension-independent bound for the Poincaré constant of every isotropic logarithmically concave distribution. Isotropic means that the mean is zero and the covariance matrix is the identity matrix. Its original geometric formulation concerns isoperimetric inequalities in convex bodies (Kannan et al. 1995).
Song and Zhang (2026) and Bizeul et al. (2026) reported proofs that an absolute constant 
satisfies 
for every isotropic logarithmically
concave distribution 
on 
and every 
.
For a general full-dimensional logarithmically
concave distribution, the equivalent bound is
 |
where 
is its covariance matrix
and 
is the operator norm. Bizeul et al. (2026)
identify a criterion of Song and Zhang (2026) as the crucial step in their proof.
Bizeul et al. (2026) state that ChatGPT found most proofs
and mathematical ideas, with the suspension idea supplied by the authors.
Balasubramanian and Kasiviswanathan (2026) reported a deterministic variational proof of the explicit bound 
for every isotropic logarithmically
concave distribution 
,
independently of dimension. Balasubramanian and Kasiviswanathan
(2026) credit AI with formulating a rank-uniform weighted Hodge comparison and proposing
Appell coefficient norms for repeated centered integration.
Zhao (2026) reports a Lean 4 formalization of the bound 
for every isotropic logarithmically
concave distribution 
,
independently of dimension. The development refines
the deterministic framework of Balasubramanian and Kasiviswanathan (2026).
Bound for the KLS Constant." 4 Oct 2026.