30-Year-Old Conjecture in Geometry Proven

In 1995, Michael Talagrand asked if convexity can be created in a fixed, uniform number of steps using Minkowski sums in any number of dimensions. In this instance, convexity is a shape in geometry or a function that bends outward, leaving no gaps or inward dents. Therefore, any line drawn from two parts on the inside of the shape would be entirely inside the shape, i.e., a circle or square in two dimensions, or a sphere or cube in three dimensions, is considered convex.

Minkowski sums are mathematical operations combining two sets of points or geometric shapes by adding every point in the first set to every point in the second set. The complication increases with each dimension. This problem is also called the “curse of dimensionality,” which causes the geometric complexity and the computation time of the resulting shapes to “explode exponentially.”

Talagrand did not believe the convexity conjecture could be solved and offered $2,000 to anyone who could provide proof. In an interview with Scientific American, he stated, “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.”

In the original 1995 paper, Talagrand showed that adding two Minkowski sets was not sufficient to guarantee the existence of a large convex subset. However, in 2025, it was proved that by replacing the Minkowski sum with convex operations, this even more complex geometric problem became false, but it did not solve Talagrand’s complex geometry problem.

Geometry Proof In Probability

The more complex geometry problem was proven by Dongming Hua and Antoine Song from the California Institute of Technology and Stefan Tudose from Princeton University. Together, they reformulated the geometric conjecture as a probability problem theory and random vectors. They published their paper on the arXiv preprint server and proved an equivalent conjecture for probability. Thus showing any one subgaussian random vector in any number of dimensions can be expressed as the sum of three Gaussian random vectors.

The result is the answer to Talagrand’s complexity problem. It proves 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.” Additionally, the solution confirms a combinatorial analog of the problem, important for discrete mathematics.

Song and Hua say they initially attempted the solution with ChatGPT. The LLM (large language model) assisted in answering some of the questions, but it was Tudose who solved the final problem. In their paper, the team states that Tudose’s proof was “more general and conceptual” and did not use the work done with ChatGPT.

The solution bridges probability and combinatorics with geometry while providing connections between discrete and continuous worlds. Even though these geometry problems seem obscure, many technologies in everyday life rely on complicated mathematics and algorithms. This solution will likely impact data science, machine learning, and logistics optimization, where complex randomness is common.

Simplified Terms

Scientific American explains it this way: draw dots on a piece of paper. Lasso all the dots within a circle. Repeat this process in any dimension. There is a known way to construct a convex shape in geometry that will contain all the points every time; however, the higher the dimension, the more difficult this procedure becomes because the shape will require more and more mathematical moves to draw.

In 1995, Talagrand believed there was an easier way to build a convex shape in geometry from high-dimensional points. In the most extreme case, a procedure of fixed complexity that does not become more difficult as the dimension grows. He believed that even in billions of dimensions, a simple shape could be constructed that encompasses the points.

Last summer, Antoine Song, a mathematician at the California Institute of Technology, translated the complex geometry question into probability theory, and it became a statement about choosing random points in space according to some rules of statistics.

Assaf Noar, a mathematician at Princeton University, says, “It was a total surprise, and I thought it was a game-changer.” When Song revealed his breakthrough during a talk at Princeton in December, Noar expected the proof to follow. “You didn’t get to the other side, but you feel like it’s going to break.”

Still, Song struggled to find the missing piece, which required manipulating a mathematical object he was not familiar with. Dongming Hua, Song’s student, turned to ChatGPT, and the LLM filled the gap in their understanding, and the proof of the proposition was revealed.

Then, Tudose, who was familiar with the object and had spent time working on his own proof, provided more valuable insight than ChatGPT. Later, the team found pre-existing publications with similar ideas to those of the chatbot. Although the information provided by AI was not used, this is the highest-profile math result that explicitly cites the use of an LLM.

Tudose says, “From my perspective, the AI didn’t change much.” This does show the mainstay of AI in the mathematician’s toolkit. “Historically, navigating unfamiliar mathematical literature required consulting specialists in the field. The advent of search engines accelerated this process, and now AI tools have made it even easier,” says Song.

Noar says, “You’re using it without knowing whenever you Google something or ask ChatGPT a question.”

Talagrand says, “I’m sure people will turn this proof in all kinds of directions. If I were 20 years younger, I would spend a year doing this to make sure I understand what is behind it.”

The mathematician won the Abel Prize in 2024. He says, “This is the most extraordinary result of my entire life. The proper word is ‘sensational.'” Until the proof appeared online last week, he did not believe his own conjecture was true.

Sources:

Phys.org: Mathematics solves decades-old mystery about the hidden order in high-dimensional randomness
Scientific American: ‘Sensational’ proof topples decades-old geometry problem

Featured Image Courtesy of Theen Moy’s Flickr Page – Creative Commons License