Sensational' proof topples decades-old geometry problem

A 30-year-old geometry conjecture about convexity in high-dimensional spaces has been proved. The conjecture claims that within extremely large, scattered, chaotic collections of points across many dimensions, orderly convex shapes must occur. Convex shapes bulge outward without dimples or crevices, meaning any straight line segment connecting two points in the shape stays entirely inside the shape. Examples include pentagons and circles, while shapes like Pac-Man are not convex. The result concerns convexity in spaces with hundreds or billions of dimensions, which matters because many computations and real datasets involve many parameters that act like dimensions.

"The conjecture says that even within enormous, scattered and chaotic assemblages of points existing across innumerable dimensions, simple, orderly shapes will inevitably crop up. French mathematician Michel Talagrand posed this convexity conjecture in 1995 as a powerful, sweeping claim about the geometry of high-dimensional shapes. He never thought he would live to see it proved. This is the most extraordinary result of my entire life, says Talagrand, who won the 2024 Abel Prize, which is often called the Nobel Prize of math."

"It’s about building convex shapes, the kind that bulge outward without any dimples or crevices. A pentagon is convex, and so is a circle, but Pac-Man isn’t: connect two points above and below his mouth with a straight line, and that line will pass beyond his yellow perimeter. For a shape to be convex, any line between two points inside of it or on its perimeter must be fully ensconced within it."

"Talagrand was interested in shapes inhabiting hundreds or billions of dimensionsor even more. This concept may seem obscure and niche, but many computations hinge on higher-dimensional math, and the real world is full of datasets with innumerable parameters that each constitute a dimension of sorts. You’re using it without knowing whenever you Google something or ask ChatGPT a question, says Assaf Noar, a mathematician at Prince"

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