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Solving a 31-year-old grand challenge in mathematics – the KLS conjecture

For decades, a fundamental question about the difficulty of computing in high dimensional spaces has resisted proof. New research resolves this question, removing a major theoretical obstacle to understanding the efficiency of randomized algorithms used across science and computing. The Kannan–Lovász–Simonovits (KLS)…

Xinzhi Zhang 6 min readOriginal

Microsoft researcher and collaborator combine new mathematical ideas with human-directed AI to resolve a central problem in high-dimensional geometry and probability

Side-by-side diagrams compare a flat cut and a curved cut through the same convex geometric shape. The left panel, labeled “Flat cut,” shows a diamond-shaped cross-section with a cut area of 0.2500. The right panel, labeled “Smallest Curved Cut,” shows a curved, ribbon-like cross-section with a smaller cut area of approximately 0.2173. Below the diagrams, text summarizes the KLS Conjecture: among cuts that divide a convex body into two equal volumes, the best flat cut has area at most a universal constant times the smallest possible cutting area in any dimension.

At a glance

  • Fundamental scientific breakthrough: Researchers resolved the Kannan-Lovász-Simonovits (KLS) conjecture, a 31-year-old problem widely regarded as one of the central open questions in high-dimensional geometry, with implications for statistics, optimization, algorithms, and computer science.
  • A universal geometric principle: The result establishes a universal geometric principle for log-concave distributions, showing that simple flat cuts can capture their narrowest bottlenecks within a fixed factor, regardless of dimension.
  • Human-AI collaboration: 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.
  • Independent confirmation: Multiple independent researchers have validated this research, identifying its central criterion as the key breakthrough, followed by a machine-checked formal verification.

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