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).
Bizeul et al. (2026) reported a proof 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)
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) acknowledge AI assistance and take responsibility
for the content. The complete argument has not been independently verified here.
