On Some Variants of the Kakeya Problem

Besicovitch and Rado [1] and Kinney [5] proved the following result: There is a closed set E ⊂ R2 with measure zero which contains a circle of every radius. The construction of Besicovitch and Rado works in Rd: If d ≥ 3 there is a closed set E ⊂ Rd with measure zero which contains a sphere of every radius. We will give an exposition of this construction in Section 1 below. One can ask whether a set containing a sphere of every radius must have Hausdorff dimension d. As it turns out, this question is easily answered in higher dimensions.