The exact $\mathbb{Z}_2$-Gromov–Hausdorff distance between Euclidean spheres of consecutive dimensions

Let $\mathbb{S}_{\mathrm{E}}^m$ denote the $m$-dimensional Euclidean sphere equipped with its standard $\mathbb{Z}_2$-action, and let $d_{\mathrm{GH}}^{\mathbb{Z}_2}$ be the $\mathbb{Z}_2$-Gromov–Hausdorff distance. For every positive integer $m$, the consecutive-sphere distance conjecture.

$$d_{\mathrm{GH}}^{\mathbb{Z}_2}(\mathbb{S}_{\mathrm{E}}^m,\mathbb{S}_{\mathrm{E}}^{m+1})=\frac{1}{2}\,\sqrt{\frac{2m+4}{m+1}}.$$

The preceding theorem establishes the matching lower bound for all positive integers $m$, while the equality is verified in the cases $m=1$ and $m=2$ by the stated exact computations. The conjecture predicts that this lower bound is sharp in every dimension.