  A K^n_2-set is a set of zero Lebesgue measure containing a translate of every
plane in an (n-2)-dimensional manifold in Gr(n,2), where the manifold fulfills
a curvature condition. We show that this is a natural class of sets with
respect to the Kakeya problem and prove that dim_H(E)\ge 7/2 for all K^4_2-sets
E. When the underlying field is replaced by the complex numbers C, we get
dim_H(E)\ge 7 for all K^4_2-sets over C, and we construct an example to show
that this is sharp. Thus K^4_2-sets over C do not necessarily have full
Hausdorff dimension.
