Define an (n,k)-Besicovitch set as a set of points in n-dimensional Euclidean space that contains a unit k-ball oritented in every direction. So, for example, a unit 3-ball (that is, a filled in sphere) in ℝ³ would be a (3,2)-Besicovitch set, since it contains 2-balls (i.e., discs) oriented in every possible direction.
The (n,k)-Besicovitch conjecture states that there are no compact (n,k)-Besicovitch sets with Lebesgue measure zero for any k>1, regardless of the value of n.
Note:
I am defining an (n,k)-Besicovitch set in the slightly different way than the linked Wikipedia article. Wikipedia includes compactness and Lebesgue nullity as part of the definition of an (n,k)-Besicovitch set (so, by Wikipedia's definition, the conjecture is just "(n,k)-Besicovitch sets don't exist for k>1".) I changed the definition because I thought that one was clunky, and because, by this new definition, an (n,k)-Besicovitch set is a proper generalization of a Besicovitch set (aka, a Kakeya set), which is just a (n,1)-Besicovitch set.
An equivalent way to define an (n,k)-Besicovitch set is that it contains translated copies of every unit k-ball centered at the origin (Wikipedia states the definition in this way).