Gromov-Hausdorff distance between interval and circle (Q2072129)

The authors introduce the new notions of round metric spaces and nonlinearity degree of a metric space. A metric space \((X, d)\) is called round if, for every \(b \in (0, \operatorname{diam} X)\) and each \(x \in X\), there exists \(y \in X\) such that \(d(x, y) \geqslant b\). The nonlinearity degree \(c(X)\) of \((X, d)\) is defined as \[ c(X) := \inf_{f \in \operatorname{Lip}_1(X)} \sup \{d(x, y) - |f(x) - f(y)| : x, y \in X\}, \] where \(\operatorname{Lip}_1(X)\) is the set of all function \(f \colon X \to \mathbb{R}\) which satisfy the inequality \(|f(x) - f(y)| \leqslant d(x, y)\) for all \(x\), \(y \in X\). Using these concepts, the authors develop an original technique enabled to obtain the exact values of the Gromov-Hausdorff distance \(d_{GH}(I_{\lambda}, S^1)\) between the interval \(I_{\lambda} = [0, \lambda] \subseteq \mathbb{R}\) and the one-dimensional sphere \(S^1 = \{z \in \mathbb{C} : |z| = 1\}\) each of which is equipped with the standard Euclidean metric, \[ d_{GH}(I_{\lambda}, S^1) = \begin{cases} \frac{\pi}{2} - \frac{\lambda}{4} & \text{if } 0 \leqslant \lambda < \frac{2}{3} \pi,\\ \frac{\pi}{3} & \text{if } \frac{2}{3} \pi \leqslant \lambda \leqslant \frac{5}{3} \pi,\\ \frac{\lambda - \pi}{2} & \text{otherwise}. \end{cases} \] The last formula is one of the few exact values of the Gromov-Hausdorff distance between given metric spaces.