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'Once in a Century' Proof Settles Math's Kakeya Conjecture
The deceptively simple Kakeya conjecture has bedeviled mathematicians for 50 years. A new proof of the conjecture in three dimensions illuminates a whole crop of related problems.
For five decades, mathematicians have sought the best possible solution to the three-dimensional version of this challenge: Hold a pencil in midair, then point it in every direction while minimizing the volume of space it moves through. Two years later, the Russian mathematician Abram Besicovitch found the answer: a complicated set of narrow turns that, counterintuitively, covers no space at all. In 2014, Larry Guth, a mathematician at the Massachusetts Institute of Technology, had proved that any counterexample to the Kakeya conjecture needed to be “grainy.” In a grainy set, there are many small 3D sections where lots of tubes overlap.
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