THE DAILY EDITION · 21 OCTOBER 2026
Researchers say they have proved the KLS conjecture
Why this story?
It settles a long-open mathematics problem that underpins sampling and randomized algorithms, with Microsoft Research named in the proof announcement.
Two researchers, including Xinzhi Zhang of Microsoft Research, say they have proved the Kannan-Lovász-Simonovits conjecture, a problem first proposed in 1995 and long treated as one of the central open questions in high-dimensional geometry. Microsoft Research said the result establishes a fixed bound for a broad class of high-dimensional probability distributions, rather than one that grows with dimension.
In its formulation, the conjecture asks whether the best flat or hyperplane cut of certain log-concave distributions is always within a universal constant of the best cut of any shape. The paper’s abstract says the authors prove the conjecture and a related bound for isotropic log-concave probability measures.
The result has implications for high-dimensional sampling, random walks on convex bodies, and algorithms used in optimization and computer science. The paper says bounds on the KLS constant control conductance and mixing times, which affect sampling, volume computation, and log-concave integration.
Microsoft Research said the conjecture had been widely regarded as one of the central open questions in high-dimensional geometry. The company statement cited prior descriptions from MSRI, the Simons Institute, and Ronen Eldan on its importance to high-dimensional convex and computational geometry, and Quanta Magazine said KLS lies at the heart of many questions in statistics and computer science.