Microsoft says researchers solved the 31-year-old KLS conjecture

Microsoft says two researchers, including Xinzhi Zhang of Microsoft Research, proved the Kannan–Lovász–Simonovits (KLS) conjecture, proposed in 1995 and considered a central problem in high-dimensional geometry. According to the company, the result establishes that flat cuts can capture the narrowest bottlenecks of log-concave distributions within a fixed factor, regardless of the number of dimensions. The conjecture has implications for probability, statistics, optimization, algorithms and high-dimensional sampling.

Microsoft says the proof emerged from a research process involving AI agents, with the systems exploring possibilities while researchers evaluated, filtered and redirected the ideas. The company says multiple independent researchers validated the work and that a machine-checked formal verification confirmed its central criterion. The post does not provide the proof’s full technical details.