Estimates of Gromov's box distance

In 1999, M. Gromov introduced the box distance function $\mathop{\underline{\square}_{\lambda}}\nolimits$ on the space of all mm-spaces. In this paper, by using the method of T. H. Colding, we estimate $\mathop{\underline{\square}_{\lambda}}\nolimits (\mathbb{S}^n,\mathbb{S}^m)$ and $\mathop{\underline{\square}_{\lambda}}\nolimits(\mathbb{C}P^n, \mathbb{C}P^m)$, where $\mathbb{S}^n$ is the $n$-dimensional unit sphere in $\mathbb{R}^{n+1}$ and $\mathbb{C}P^n$ 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 $\mathop{\underline{\square}_{\lambda}}\nolimits\big(SO(n), SO(m)\big)$ from below, where $SO(n)$ is the special orthogonal group.