Kakeya problem and projection problem for k-geodesics in Grassmannians

The Kakeya problem in Rⁿ is about estimating the size of union of k-planes; the projection problem in Rⁿ is about estimating the size of projection of a set onto every k-plane (1≤ k≤ n-1). The $k=1$ case has been studied on general manifolds in which $1$-planes become geodesics, while k≥ 2 cases were still only considered in Rⁿ. We formulate these problems on homogeneous spaces, where k-planes are replaced by k-dimensional totally geodesic submanifolds. After formulating the problem, we prove a sharp estimate for Grassmannians.