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
Note:
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).