Best constants for metric space inversion inequalities (Q352755)

Let \((X,d)\) be a metric space with \(\text{card}(X)\geq 2\). Given some distinguished point \(0\in X\) (origin), denote \[ I_0(x,y)=\frac{d(x,y)}{d(x,0)d(y,0)}, \qquad x,y\in X_0, \] the metric space inversion semimetric, and let \(d_0(\cdot,\cdot)\) be its subordinated metric. Then, for \(x,y\in X_0\), let \(\widehat d_0(x,y)\) be the infimum of distances \((d_Y)^{}_0(x,y)\) over all metric spaces \((Y,d_Y)\) containing an isometric copy of \(X\). Finally, let \(\widehat C_{\mathrm{inv}}(X)\) be the smallest constant \(C\geq 1\) such that (for each origin \(0\in X\)) \(I_0(x,y)\leq C{\widehat d_0}(x,y)\) for all \( x,y\in X_0\). Theorem. The following assertions hold: i) \(\widehat C_{\mathrm{inv}}(X)\leq 2\); ii) \(\widehat C_{\mathrm{inv}}(X)=2\) if and only if, for each \(\varepsilon> 0\), there exists an origin \(0\in X\) and two points \(x,y\in X_0\) such that \[ \frac{d(x,y)}{2d(x,0)}> 1-\varepsilon\quad\text{and}\quad \frac{d(x,y)}{2d(y,0)}> 1-\varepsilon. \] Some related contributions in the area are then discussed under this perspective.