On the existence of bi-Lipschitz equivalent metrics in semimetric spaces (Q6088280)

The paper revisits certain known results on the existence of a bi-Lipschitz equivalent metric with respect to a given semimetric space, quasi-ultrametric space and quasi-metric space. The existence of a bi-Lipschitz equivalent metric is given in terms of relaxed polygonal inequality (RPI) (Theorem 2, Theorem 4, Theorem 5). Here note that a semimetric \(\rho\) on a nonempty set \(X\) satisfies the relaxed polygonal inequality with constant \(C > 0\) if for any \(n \in \mathbb{N}\) and for any \(\{x_o, x_1,\ldots, x_n\} \subseteq X\) with \(x_o = x\) and \(x_n = y\), the following condition holds: \[ \rho(x,y) \leq C \sum \limits_{i=1}^n \rho(x_{i-1}, x_i). \]