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Can we compute the Gromov-Hausdorff distance $d(\mathbb{S}_n,\mathbb{S}_m)$ for two different spheres $\mathbb{S}_n$ and $\mathbb{S}_m$, $m\neq n$? We consider the spheres with the metrics induced by the embedding $\mathbb{S}_n \to \mathbb{R}^{n+1}$.

For $n=2$, $m=3$, this is evaluating $$d(\mathbb{S}_2,\mathbb{S}_3)=\inf_{M,f,g}d_{M}(\mathbb{S}_2,\mathbb{S}_3)$$ where $M$ ranges over all possible metric spaces and $f:\mathbb{S}_2\to M$ and $g:\mathbb{S}_3\to M$ range over all possible distance-preserving embeddings. As an upper bound, $d(\mathbb{S}_2,\mathbb{S}_3)\leq \sqrt{2}$.

More generally for $0 \leq n \leq m$, $$d(\mathbb{S}_n,\mathbb{S}_m)\leq d(point,S_n)+d(point,S_m)\leq 2$$ But I find it difficult to control the lower bound with the inf over all possible metric spaces $M$.

I conjecture that for all $0 \leq n \leq m$: $$d(\mathbb{S}_n,\mathbb{S}_m)\geq \lambda_{m,n}\frac{m-n}{m}, \text{ where } \liminf_{m,n\to \infty}\lambda_{m,n}>0$$

I only know the Gromov-Hausdorff theory from Petersen's Riemannian Geometry, which does not give enough information to compute this distance. I will appreciate any pointers.

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    $\begingroup$ Please check your question for typos. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Oct 28, 2017 at 3:58
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    $\begingroup$ @Mark Sapir,Appreciate for help!I am reading the article you point out,it seems this article mainly focus on investigating the Gromov-Hausdorff limit space of a sequence of hyperbolic group equipped with modified G-H metric defined in 2.A with some special condition to ensure the limit space exists.and take a sequences corvarage to the limit space,the hyperbolic property and some other thing is stayed by the process of take limit. $\endgroup$
    – Hu xiyu
    Commented Oct 28, 2017 at 4:26
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    $\begingroup$ Your conjecture would imply that the GH distance is unbounded. But it's clearly bounded, since the GH distance of any sphere to a point is equal to 2 (when the sphere is endowed with the restriction of Euclidean distance, as you seem to assume, or $\pi$ when endowed with geodesic distance) and hence the GH distance between any two spheres is $\le 4$. $\endgroup$
    – YCor
    Commented Oct 28, 2017 at 7:29
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    $\begingroup$ Clearly from standard embeddings we get $d_{GH}(S_n,S_m)\le\sqrt{2}$ for all $n,m\ge 0$. Would it be reasonable to simply conjecture that it's an equality whenever $n\neq m$? $\endgroup$
    – YCor
    Commented Oct 28, 2017 at 7:59
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    $\begingroup$ And in addition there is the obvious remark (I didn't first notice) that $d(S_n,S_m)\le 1$ for all $n,m$ (form a metric space $S_n\sqcup S_m$ with the given distance on each component and points in different components being at distance 1. This works for any two metric spaces of diameter $\le 2$. So my previous comment should rather ask whether $d_{GH}(S_n,S_m)=1$ for all $n\neq m$. $\endgroup$
    – YCor
    Commented Oct 28, 2017 at 10:08

1 Answer 1

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Hu Xiyu. Even though yours is a question from four years ago, I want to bring to your attention my recent paper “Gromov-Hausdorff distance between spheres”( https://arxiv.org/abs/2105.00611 ) coauthored with Facundo Memoli and Zane Smith, since it is very closely related to your question.

In this paper, we compute/bound the Gromov-Hausdorff distance between two spheres with different dimension (each with geodesic metric). We use topological methods in order to obtain lower bounds: more precisely, we resort to a certain version of Borsuk-Ulam Theorem for discontinous functions. On the other hand, we design specialized optimal correspondences in order to estimate upper bounds. In particular, we were able to compute precise value of the Gromov Hausdorff distance for $\mathbb{S}^1$ vs $\mathbb{S}^2$, $\mathbb{S}^1$ vs $\mathbb{S}^3$, and $\mathbb{S}^2$ vs $\mathbb{S}^3$.

Finally, the last section of the paper deals with the case of spheres with Euclidean metric. Even though we could not give a full answer, I believe you will be able to find some useful observations there.

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