From the source
Microsoft researchers and a collaborator have proved the Kannan-Lovász-Simonovits conjecture, a 31-year-old open problem in high-dimensional geometry, showing the bound holds with a fixed constant regardless of dimension.
- The KLS conjecture, proposed in 1995, asks whether the best flat or hyperplane cut of a broad class of high-dimensional distributions is nearly as effective as the best cut of any shape, differing by no more than a universal constant.
- Before this work, the best known result, established in July 2026, allowed that bound to grow slowly as dimensions increased.
- The authors improved that result using a human-directed agentic process in which AI agents explored more than 100 candidate approaches, many involving lengthy calculations with higher-order tensors, while the researchers evaluated, pruned and redirected the search.
- A new criterion was applied recursively, and in the October 1, 2026 version of the paper the dependence on dimension fell to the iterated logarithm, which still grows with dimension.
- Repeating the entire recursive refinement, an idea inspired by the authors' earlier work on fast linear programming algorithms, showed the accumulated errors stay bounded by a universal constant, completing the proof in the October 4, 2026 version.
- On the same day, a separate group of leading experts posted a full proof developed in parallel that built on the first version of the paper, and a third proof relying on a closely related idea appeared separately.
