Mathematicians presented a solution to a problem they couldn't prove for 30 years

The problem sounded very abstract: whether in spaces with a huge number of dimensions it is possible to find a sufficiently "even" structure within a complex random set in a limited number of steps. Simply put, it is about whether there is a hidden order where at first glance everything looks too complex and random.

The new proof was presented by Dongmin Merrick Hua, Antoine Song and Stephan Tudose. The paper is published as a preprint on arXiv, so it is more correct to say that the mathematicians have stated the solution: the proof must now be tested by the scientific community.

The details

To understand the idea, we can start with a simple example. A convex figure is a figure without "depressions". If you take two points inside a circle or square and connect them with a line, the whole line will also stay inside the figure. That is convexity.

But Talagran was not interested in ordinary circles and squares, but in much more complex objects - sets in spaces with any number of dimensions. In such spaces, the usual geometric intuition almost does not work: the more dimensions, the more difficult it is to understand how the shape is organised.

Talagran's question was something like this: if we have a large set in a high-dimensional space, can we use a fixed number of operations to get a large enough convex part inside it? Importantly, the number of steps should not grow with the number of dimensions.

The authors of the new proof approach the problem not only as a geometric problem, but also as a probability problem. In a preprint, they formulate the result in terms of random vectors and show that this solves the Talagran convexity problem and also gives a corollary for a similar problem in combinatorics.

Why it matters

Such problems may sound far removed from ordinary life, but they are at the heart of the maths that helps us understand complex data. High-dimensional spaces appear in statistics, data analysis, machine learning and optimisation: for example, when an object is described by thousands of parameters rather than two or three features.

This doesn't mean that a proof will change the way neural networks work tomorrow or provide a new algorithm for business. This is fundamental maths. But results like this are gradually changing the tools scientists use to describe complex random systems.

The main value of the paper is that it shows: even in very complex randomness, there can be structure that can be described by rigorous mathematical methods.

Background

Michel Talagran is known for his work in probability theory, functional analysis and high dimensional geometry. He formulated the convexity problem itself in 1995. According to Scientific American's paraphrase, Talagran called his conjecture almost a "shot in the dark" and wasn't sure if it was actually true.

An interesting detail is the role of AI. According to reports on the paper, early on the authors used ChatGPT as an assistant to discuss individual ideas, but the final proof was generated by mathematicians, not the model.

Source

Dongming Merrick Hua, Antoine Song, Stefan Tudose, "On Talagrand's Convexity Conjecture," arXiv, 2026.

In the paper, the authors prove a result on random vectors which, according to their formulation, solves Talagrand's convexity problem and gives a consequence for its combinatorial analogue. Since the paper is still posted as a preprint, the conclusions should be presented with a caveat: the proof is presented, but it will still be checked by experts.