Here is a maths puzzle that is easy to picture and very hard to prove. Imagine a cloud of points in space, and you want to cut it into two roughly equal halves. You could cut it with a flat slice, like a knife through a cake, or with a wiggly, curved cut of any shape you like. The curved cut might be a little better — but how much better could it possibly be?
In 1995, mathematicians Ravi Kannan, Laszlo Lovasz and Miklos Simonovits asked a precise version of that question for a broad class of high-dimensional distributions — the kind of "clouds" that appear in statistics, optimisation and computer science. Their conjecture, known as KLS, asked whether the best flat cut is always nearly as good as the best cut of any shape, differing by no more than a fixed, universal amount that does not grow as you add more dimensions.
For 31 years, the answer kept getting closer but never quite arrived. Each new result shrank how much the bound grew with dimension, but the bound still grew. In July 2026, the best known result still allowed slow growth with dimension.
On 21 October 2026, Microsoft Research announced that senior researcher Xinzhi Zhang and collaborator Zhao Song have proved the conjecture. Their paper, "An O(1) Bound for the KLS Constant", was first posted on arXiv on 1 October 2026 and revised on 4 October 2026. In plain terms, the "O(1)" in the title means the bound is now a fixed constant: it does not grow with dimension at all.
Microsoft also described how the work was done. AI agents, directed by the human researchers, explored more than 100 candidate approaches, while the researchers evaluated, pruned and redirected the search. Independent groups also released their own proofs around the same time, and a formal verification in the Lean 4 system has been reported. That matters because a result this important will be checked, re-checked and built upon — an announced proof is the start of that process, not the end of it.
Why should school students care about a problem in many dimensions? Because it shows what maths really is. It is not just calculating answers; it is asking a question so clearly that it can survive 31 years of attack, and then finding a way to answer it. Dimensions are not just science-fiction either: every extra piece of information about something — its height, its weight, its price, its age — is another dimension in the data. Understanding how to cut, sort and sample high-dimensional data is part of how modern statistics and computing work.
A useful way to try the idea at home: draw a blob on paper, and try to split it into two equal areas with one straight line. Then try with a wiggly line. How much better can the wiggly line really do? That small experiment is the two-dimensional version of a question that took three decades to settle in general.
Sources:
- Microsoft Research announcement, 21 October 2026, on the KLS conjecture proof by Xinzhi Zhang and Zhao Song, as reported by Unite.AI: https://www.unite.ai/microsoft-researcher-and-collaborator-prove-31-year-old-kls-conjecture/
- Brief, "31-Year-Old KLS Conjecture Solved: Breakthrough in High-Dimensional Geometry and Probability", 21 October 2026: https://www.brief.news/ai-research/2026/10/21/geometric-breakthrough-solves-kls-conjecture?briefFormat=TopStories&categories=Science&categories=World&categories=US&categories=AI&categories=Gaming&categories=Tech&categories=Financial+Markets&categories=Space&source=%2Fai&utm_source=category-page&utm_medium=web
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