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Is the Kakeya conjecture true?
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A Kakeya set (also called a Besicovitch set) is a set in Euclidean space that contains a unit line segment pointing in every direction.
- The Kakeya conjecture states that a Kakeya set in ℝⁿ always has a Hausdorff dimension of n.
- For more information, see the Wikipedia article: https://en.wikipedia.org/wiki/Kakeya_set
- You don't need to be able to continuously rotate a line segment through all possible orientations in a Kakeya set. a set where you can also do that for a segment of length 1 is called a Kakeya needle set and is irrelevant here (The conjecture is trivially true for all Kakeya needle sets, since they must have positive area).
- The definition of the Kakeya set conjecture given in the Wikipedia article above also says that the Minkowski dimension has to be equal to n.
- This is actually equivalent to my definition because Hausdorff dimension ≤ Minkowski dimension ≤ n for all subsets of Euclidean n-space for which the dimensions are all defined, and the Minkowski dimension is defined for any set with Hausdorff dimension n (since any such set must have upper and lower box dimension equal to n, meaning the Minkowski dimension is n).