Microsoft researcher helps prove 31-year-old geometry conjecture

Since 1995, one question about high-dimensional geometry has stood between mathematicians and a clean answer about how random walks explore a distribution. The Kannan–Lovász–Simonovits conjecture asks whether, for a broad class of probability distributions, the best flat cut is nearly as effective as the best cut of any shape. Microsoft Research now reports that two researchers, including Xinzhi Zhang of its Machine Learning and Optimization group, have proved it: the gap between the two kinds of cut stays below a fixed bound, whatever the number of dimensions.

A flat cut is a slice through a shape made with a flat blade, a hyperplane in mathematical terms. For log-concave distributions, a broad family of probability distributions, the conjecture predicts that the best flat cut differs from the best cut of any shape by no more than a universal constant, however many dimensions there are. The geometry of these cuts sets how quickly a random walk can explore the distribution, which is why the problem sits at the centre of high-dimensional sampling, the family of methods that draw random points from complicated distributions.

The problem was recognised as central long before it fell. The Mathematical Sciences Research Institute called it “a central open problem in high-dimensional convex geometry” in its 2017-18 Annual Report, and the Simons Institute described it as “one of the most important open problems in convex geometry.” The best known result, established in July 2026, let the bound grow slowly as dimensions increased. The authors first sharpened it: in the version posted on October 1, 2026, the dependence on dimension fell to the iterated logarithm, a function that grows very slowly but still grows, so the conjecture was not yet resolved.

The last steps came from an unusual workflow. Working with AI agents in a structured, human-directed process, the authors explored more than 100 candidate approaches, many involving lengthy calculations with higher-order tensors. One of them produced a new criterion that became the foundation of the proof. The final move was to repeat the whole recursive refinement rather than apply it once, an idea inspired by the authors’ earlier work on fast algorithms for linear programming. They showed that the small errors introduced at each stage shrink fast enough that their total stays bounded by a universal constant, independent of dimension. The version posted on October 4, 2026 completed the proof. The authors say AI tools helped explore, derive and check, while the researchers chose the direction, selected the ideas and verified the argument.

The claim has drawn parallel work. On the same day the Microsoft team posted its complete proof, a separate group of leading experts posted a full proof it had developed in parallel, calling the first Microsoft version “the crucial step in the proof.” A third proof, posted that day, relies on a closely related idea. The Microsoft post also reports that several independent researchers validated the work and that a machine-checked formal verification followed. It does not say whether the paper has yet been accepted by a journal.

For researchers who build and analyse sampling and optimisation methods, the change is in the guarantees: a bound that no longer grows with dimension replaces one that grew, however slowly. The post presents this as the removal of a major theoretical obstacle to understanding the efficiency of randomized algorithms used across science and computing. It sets no date for any practical effect, and it does not say whether the result will translate into faster tools.