May 24, 2026 [ report
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Krystal Kasal
Author
Gaby Clark
Scientific Editor
Robert Egan
Associate Editor
Three mathematicians have laid out proof that solves a long-standing problem in mathematics. Even the mathematician—an Abel prize winner—that first posed the problem didn't believe it would ever be solved. The solution provides insight into high-dimensional random structures that could potentially impact data science, machine learning and optimization.
Talagrand's convexity conjecture #
In 1995, Michel Talagrand came up with his famous mathematical problem, which asks whether convexity can be "created" in a fixed, uniform number of steps (using operations called Minkowski sums) in any number of dimensions. In mathematics, convexity means that a shape or function bends outward, ensuring no gaps or inward dents exist. So, any line drawn from two points on the perimeter or inside of the shape should lie entirely within the shape. For example, a circle or square in two dimensions, or a sphere or cube in three dimensions, would be considered convex.
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Talagrand's convexity conjecture requires Minkowski sums, which are mathematical operations that combine two sets of points or geometric shapes by adding every single point in the first set to every point in the second set. All of this gets more complicated as the number of dimensions increases. Some refer to this problem as the "curse of dimensionality," which causes both the geometric complexity and the computation time of the resulting shapes to explode exponentially.
Talagrand himself didn't think the convexity conjecture was solvable and offered $2,000 to anyone who could come up with the proof. He told Scientific American, "I made this bold conjecture really without any ground for it, you know—it's just a shot in the dark. When you say something like that, you feel it cannot possibly be true."
Talagrand originally showed in his 1995 paper that two Minkowski additions are not enough to guarantee the creation of a large convex subset. In 2025, another mathematician proved that replacing the Minkowski sum with convex operations makes this stronger version of the convexity problem false. But this still didn't solve Talagrand's more general version.
Finding proof in probability #
The new proof was worked out by Dongming Hua and Antoine Song from the California Institute of Technology, and Stefan Tudose from Princeton University, who joined the other authors after hearing about their work. Together, the mathematicians reformulated Talagrand's geometric conjecture to a problem of probability theory and random vectors. In their paper published on the arXiv preprint server, they proved an equivalent conjecture for probability, showing that any 1-subgaussian random vector in n dimensions can be expressed as the sum of three standard Gaussian random vectors.
This result solves Talagrand's convexity problem, proving that for any large enough set in Gaussian space, a convex set of significant measures can be found inside a triple sum of the original set. The solution also confirms a combinatorial analog of the problem, which is important for discrete mathematics.
Initially, Song and Hua say they attempted to work out a solution with the help of ChatGPT. However, while the LLM helped to answer some of their questions and move them closer to a solution, it was Tudose who provided the final proof. Ultimately, the team did not use the work done with ChatGPT. In their paper, the team writes that Tudose's proof was "more general and conceptual."
The solution to this decades-old mathematical mystery bridges geometry, probability and combinatorics, and provides some surprising connections between continuous and discrete worlds. Although these kinds of mathematical problems may feel obscure, many technologies involved in our everyday lives rely on complicated mathematical tools and algorithms. The solution to Talagrand's conjecture may impact data science, machine learning and things like logistics optimization, where similar models involving complex randomness are common.
Written for you by our author Krystal Kasal, edited by Gaby Clark, and fact-checked and reviewed by Robert Egan—this article is the result of careful human work. We rely on readers like you to keep independent science journalism alive.
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Publication details
Dongming Merrick Hua et al, On Talagrand's Convexity Conjecture, arXiv (2026). DOI: 10.48550/arxiv.2605.10908 Journal information:
[arXiv](https://phys.org/journals/arxiv/)
[
](http://arxiv.org/)
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Citation: Mathematicians solve decades-old mystery about the hidden order in high-dimensional randomness (2026, May 24) retrieved 25 May 2026 from https://phys.org/news/2026-05-mathematicians-decades-mystery-hidden-high.html