Estimates of Gromov's box distance

In 1999, M. Gromov introduced the box distance function □λ on the space of all mm-spaces. In this paper, by using the method of T. H. Colding, we estimate □λ(Sⁿ,S^m) and □λ(ℂ Pⁿ,ℂP^m), where Sⁿ is the n-dimensional unit sphere in ℝ^{n +1} and ℂPⁿ is the n-dimensional complex projective space equipped with the Fubini-Study metric. In particular, we give the complete answer to an exercise of Gromov's green book. We also estimate □λ (SO(n),SO(m)) from below, where SO(n) is the special orthogonal group.