Some novel constructions of optimal Gromov-Hausdorff-optimal correspondences between spheres

TL;DR

This work advances the exact computation of the Gromov-Hausdorff distance between spheres with geodesic metrics by constructing explicit metric correspondences. It provides alternative proofs of the sharp bounds for $d_{GH}(bS^1,bS^{2n})$ and $d_{GH}(bS^1,bS^{2n+1})$, both equalling $\frac{\pi n}{2n+1}$, and resolves the $n=3$ case of the Lim–Mémoli–Smith conjecture by showing $d_{GH}(bS^3,bS^4)=\frac{1}{2}\arccos(-\tfrac{1}{4})$. The methods rely on metric correspondences derived from helmet-trick reductions, regular simplices, and carefully designed interpolations of maps (notably $\Phi$ and $F_n$) to tightly control distortion. The results contribute to a precise understanding of how round spheres with different dimensions fail to be isometric under GH distance and highlight the role of structured geometric constructions in obtaining sharp bounds. The paper also discusses computer-assisted approaches to verify intricate inequalities that arise in higher-dimensional constructions.

Abstract

In this article, as a first contribution, we provide alternative proofs of recent results by Harrison and Jeffs which determine the precise value of the Gromov-Hausdorff (GH) distance between the circle $\mathbb{S}^1$ and the $n$-dimensional sphere $\mathbb{S}^n$ (for any $n\in\mathbb{N}$) when endowed with their respective geodesic metrics. Additionally, we prove that the GH distance between $\mathbb{S}^3$ and $\mathbb{S}^4$ is equal to $\frac{1}{2}\arccos\left(\frac{-1}{4}\right)$, thus settling the case $n=3$ of a conjecture by Lim, Mémoli and Smith.

Figures (12)

• Figure 1: A low dimensional depiction of the maps $F'$ (left) and $F"$ (right).
• Figure 2: The Voronoi cells $V_i^{2n}$ and $W_i$ in the case $n=1$
• Figure 3: Some subsets of $\mathbb{S}^{2n+1}$ we will use in our construction.
• Figure 4: The map $\Phi:D\to\mathbb{S}^{1}$. This figure depicts the restriction of $\Phi$ to $\mathbb{S}^{2n}=\mathbb{S}^2$ and also its restriction to two geodesics (colored blue) between points of $\mathbb{S}^{2n}$ and $N$, the north pole of $\mathbb{S}^3$. We utilize two copies of $N$ for more clarity. Note that all points in the geodesic segment $[p',N]$ with $\alpha>\frac{\pi}{2n+1}=\frac{\pi}{3}$ are mapped to $-q_0\cdot e^{\frac{i\pi}{2n+1}}=q_2$.
• Figure 5: The image of $\Phi$ is half of $\mathbb{S}^1$ (case $n=2$)

Theorems & Definitions (51)

• Example 1.1
• Conjecture 1.2: LMS
• Theorem 1.3: HJ
• Theorem 1.4: HJ
• Theorem 1.5
• Example 1.6
• Remark 1.7
• Remark 1.9: Computer assisted proofs of inequalities
• Remark 1.10
• Proposition 2.2: Helmet trick for relations, cf. LMS