New geodesic lines in the Gromov-Hausdorff class lying in the cloud of the real line
Ivan N. Mikhailov
TL;DR
The work investigates geodesic structure in the Gromov–Hausdorff space and its clouds by establishing a fundamental distance lower bound for l^1-product spaces. It proves that for an unbounded subset $A\subset \mathbb{R}$ and a bounded metric space $X$, the curve $A \times_{\ell^1} (tX)$ forms a GH-geodesic as $t$ varies, highlighting a constructive route to geodesics in the cloud of the real line. The authors then construct a concrete obstruction to geodesics in higher-dimensional clouds by showing $d_{GH}(\mathbb{Z}^n, \lambda\mathbb{Z}^n) \ge \tfrac{1}{2}$ for $\lambda>1$, implying that the curve $t\mathbb{Z}^n$ is not continuous in GH distance and thus not a geodesic; this also demonstrates that scaling all spaces in the cloud $[\mathbb{R}^n]$ by $\lambda$ is not GH-continuous. Collectively, these results reveal both a viable geodesic construction in the real-line cloud and intrinsic discontinuities in higher-dimensional clouds, informing the limits of contractibility arguments in the GH framework.
Abstract
In the paper we prove that, for arbitrary unbounded subset $A\subset R$ and an arbitrary bounded metric space~$X$, a curve $A\times_{\ell^1} (tX)$, $t\in[0,\,\infty)$ is a geodesic line in the Gromov--Hausdorff class. We also show that, for abitrary $λ> 1$, $n\in\mathbb{N}$, the following inequality holds: $d_{GH}\bigl(\mathbb{Z}^n,\,λ\mathbb{Z}^n\bigr)\ge\frac{1}{2}$. We conclude that a curve $t\mathbb{Z}^n$, $t\in(0,\,\infty)$ is not continuous with respect to the Gromov--Hausdorff distance, and, therefore, is not a gedesic line. Moreover, it follows that multiplication of all metric spaces lying on the finite Gromov--Hausdorff distance from $\mathbb{R}^n$ on some~$λ> 0$ is also discontinous with respect to the Gromov--Hausdorff distance.