AI 中文总结
该研究解决了Lim等人关于连续球面格罗莫夫-豪斯多夫距离的猜想，构造对应关系族并引入同步球面连接与悬置操作，得到非连续维球面的距离下界。
AI 中文摘要
我们确定了配备测地距离的连续单位正则球面之间的格罗莫夫-豪斯多夫距离。令ζₙ := arccos(-1/(n+1))，这是内接于Sⁿ的n+2个顶点的正则单纯形的不同顶点之间的公共测地距离。我们证明了d_GH(Sⁿ, Sⁿ⁺¹) = ζₙ/2（n≥1），解决了Lim、Mémoli和Smith提出的猜想，此前n≥4的所有情况均未解决。该等式通过显式构造对应关系族Rₙ⊆Sⁿ⁺¹×Sⁿ来建立，其失真与已知的定量博苏克-乌拉姆下界ζₙ匹配。我们还引入了同步球面连接与对应关系的悬置，并证明连接的失真恰好是其各因子失真的最大值，特别地，悬置保持失真。将这些连接与悬置操作应用于最优对应关系Rₙ，可得到非连续维球面的新下界，包括当d(m)≥1且d(m)=o(m)时，lim_{m→∞} d_GH(Sᵐ, Sᵐ⁺ᵈ⁽ᵐ⁾) = π/4。
英文摘要
We determine the Gromov-Hausdorff distance between consecutive unit round spheres equipped with their geodesic metrics. Put $ζ_n:=\arccos(-\tfrac{1}{n+1}),$ the common geodesic distance between distinct vertices of a regular simplex with $n+2$ vertices inscribed in $\mathbb{S}^n$. We prove that $$ d_{\mathrm{GH}}(\mathbb{S}^n,\mathbb{S}^{n+1})=\frac{ζ_n}{2} \qquad(n\geq1), $$ resolving a conjecture of Lim, Mémoli, and Smith. All cases $n\geq4$ were previously open. This equality is established by explicitly constructing a family of correspondences $\mathcal R_n\subseteq \mathbb{S}^{n+1}\times \mathbb{S}^n$, whose distortion matches the known quantitative Borsuk-Ulam lower bound $ζ_n$. We also introduce synchronized spherical joins and suspensions of correspondences and prove that the distortion of a join is exactly the maximum of the distortions of its factors. In particular, suspension preserves distortion. Applying these join and suspension operations to the optimal correspondences $\mathcal R_n$ yields new bounds for spheres of nonconsecutive dimensions, including $$ \lim_{m\to\infty} d_{\mathrm{GH}}\bigl(\mathbb{S}^m,\mathbb{S}^{m+d(m)}\bigr) = \fracπ{4} \qquad\text{whenever } d(m)\geq1,\ \text{and }d(m)=o(m).$$
