Estimates of Gromov's box distance

Abstract

In 1999, M. Gromov introduced the box distance function $\sikaku$ on the space of all mm-spaces. In this paper, by using the method of T. H. Colding (cf. \cite[Lemma 5.10]{Colding}), we estimate $\sikaku(\mathbb{S}^n,\mathbb{S}^m)$ and $\sikaku (\mathbb{C}P^n, \mathbb{C}P^m)$, 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 paticular, we give the complete answer to an Exercise of Gromov's Green book (cf. \cite[Section $31/2.18$]{gromov}). We also estimate $\sikaku \big(SO(n), SO(m)\big)$ from below, where SO(n) is the special orthogonal group.

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