Microsoft Research team proves 31-year-old KLS conjecture in high-dimensional geometry

A team at Microsoft Research announced the resolution of the Kannan–Lovász–Simonovits (KLS) conjecture, a 31-year-old problem in high-dimensional geometry that had resisted proof since its proposal in 1995. The conjecture asks whether, for a broad class of high-dimensional probability distributions, the best flat or hyperplane cut is nearly as effective as the best cut of any shape, with a universal constant bound independent of dimensionality. The result establishes that such a dimension-free geometric principle holds for log-concave distributions, removing a major theoretical obstacle to understanding the efficiency of randomized algorithms used across science and computing.

The breakthrough emerged from a human-directed, agentic research process in which AI systems explored mathematical possibilities while researchers evaluated, pruned, and redirected the search. The authors reported exploring more than 100 candidate approaches, many involving lengthy calculations with higher-order tensors, before identifying a new mathematical criterion that became the foundation of the final proof. This criterion was then applied recursively, with each improved bound fed back into the argument to produce a tighter one, ultimately reducing the dependence on dimension to the iterated logarithm in an October 1, 2026 version of the paper.

The final step involved repeating the recursive refinement process to eliminate the remaining dependence on dimensionality. The authors proved that the small errors introduced at each stage shrink fast enough that their total remains bounded by a universal constant, independent of the number of dimensions. The recursive method, its repeated application, and the proof that accumulated errors remain bounded were identified as original contributions of the researchers. AI tools assisted with exploring candidate approaches, carrying out detailed derivations, preparing the manuscript, and performing internal proof checks, but the researchers determined the mathematical direction, selected promising ideas, assembled the final argument, and verified the proof.

Independent confirmation of the result arrived on the same day the authors posted their complete proof. A separate group of leading experts posted a full proof developed in parallel, calling the Microsoft team’s initial findings “the crucial step in the proof.” A third proof, appearing separately the same day, relied on a closely related idea. Additionally, a Lean 4 formalization of the proof was subsequently released, providing a machine-checked verification of the result.

The KLS conjecture’s importance was underscored by its inclusion in reports from the Mathematical Sciences Research Institute (MSRI) and the Simons Institute, which described it as a central open problem in high-dimensional convex geometry. Coverage by Quanta Magazine emphasized its broad implications for statistics and computer science, noting that the conjecture “lies at the heart of many questions in statistics and computer science.” The resolution establishes a universal geometric principle that clarifies why high-dimensional spaces can be navigated efficiently, with consequences for the theory of high-dimensional sampling and the design of algorithms.